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Title: A History of Mathematics 

Author: Florian Cajori 

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Formerly Professor of Applied Mathematics in the Tulane University 

of Louisiana; now Professor of Physics 

in Colorado College 

"I am sure that no subject loses more than mathematics 
by any attempt to dissociate it from its history." — J. W. L. 

Beta fork 




All rights reserved 

Copyright, 1893, 

Set up and electrotyped January, 1894. Reprinted March, 
1895; October, 1897; November, 1901; January, 1906; July, 1909. 

JiottoooB $re|T: 

J. S. Cushing & Co. — Berwick & Smith. 
Norwood, Mass., U.S.A. 


An increased interest in the history of the exact sciences 
manifested in recent years by teachers everywhere, and the 
attention given to historical inquiry in the mathematical 
class-rooms and seminaries of our leading universities, cause 
me to believe that a brief general History of Mathematics will 
be found acceptable to teachers and students. 

The pages treating — necessarily in a very condensed form — 
of the progress made during the present century, are put forth 
with great diffidence, although I have spent much time in 
the effort to render them accurate and reasonably complete. 
Many valuable suggestions and criticisms on the chapter on 
"Recent Times" have been made by Dr. E. W. Davis, of the 
University of Nebraska. The proof-sheets of this chapter have 
also been submitted to Dr. J. E. Davies and Professor C. A. 
Van Velzer, both of the University of Wisconsin; to Dr. G. B. 
Halsted, of the University of Texas; Professor L. M. Hoskins, of 
the Leland Stanford Jr. University; and Professor G. D. Olds, 
of Amherst College, — all of whom have afforded valuable 
assistance. I am specially indebted to Professor F. H. Loud, of 
Colorado College, who has read the proof-sheets throughout. 
To all the gentlemen above named, as well as to Dr. Carlo 
Veneziani of Salt Lake City, who read the first part of my work 
in manuscript, I desire to express my hearty thanks. But in 
acknowledging their kindness, I trust that I shall not seem to 


lay upon them any share in the responsibility for errors which 
I may have introduced in subsequent revision of the text. 


Colorado College, December, 1893. 





The Babylonians 5 

The Egyptians 10 

The Greeks 17 

Greek Geometry 17 

The Ionic School 19 

The School of Pythagoras 22 

The Sophist School 26 

The Platonic School 33 

The First Alexandrian School 39 

The Second Alexandrian School 62 

Greek Arithmetic 72 

The Romans 89 


The Hindoos 97 

The Arabs 116 

Europe During the Middle Ages 135 

Introduction of Roman Mathematics 136 

Translation of Arabic Manuscripts 144 

The First Awakening and its Sequel 148 


The Renaissance 161 

Vieta to Descartes 181 

Descartes to Newton 213 

Newton to Euler 231 




Euler, Lagrange, and Laplace 286 

The Origin of Modern Geometry 332 


Synthetic Geometry 341 

Analytic Geometry 358 

Algebra 367 

Analysis 386 

Theory of Functions 405 

Theory of Numbers 422 

Applied Mathematics 435 


The following books, pamphlets, and articles have been used in 
the preparation of this history. Reference to any of them is made 
in the text by giving the respective number. Histories marked 
with a star are the only ones of which extensive use has been 

1. GtJNTHER, S. Ziele und Resultate der neueren Mathematisch- 

historischen Forschung. Erlangen, 1876. 

2. Cajori, F. The Teaching and History of Mathematics in the U. S. 

Washington, 1890. 

3. *Cantor, Moritz. VorlesungenuberGeschichtederMathematik. 

Leipzig. Bd. I., 1880; Bd. II., 1892. 

4. Epping, J. Astronomisches aus Babylon. Unter Mitwirkung von 

P. J. R. Strassmaier. Freiburg, 1889. 

5. Bretschneider, C. A. Die Geometrie und die Geometer vor 

Euklides. Leipzig, 1870. 

6. *Gow, James. A Short History of Greek Mathematics. Cambridge, 


7. *Hankel, Hermann. Zur Geschichte der Mathematik im 

Alterthum und Mittelalter. Leipzig, 1874. 

8. *Allman, G. J. Greek Geometry from, Thales to Euclid. Dublin, 


9. De Morgan, A. "Euclides" in Smith's Dictionary of Greek and 

Roman Biography and Mythology. 

10. Hankel, Hermann. Theorie der Complexen Zahlensysteme. 

Leipzig, 1867. 

11. Whewell, William. History of the Inductive Sciences. 

12. Zeuthen, H. G. Die Lehre von den Kegelschnitten im Alterthum. 

^Copenhagen, 1886. 



13. *Chasles, M. Geschichte der Geometric Aus dem Franzosischen 

iibertragen durch Dr. L. A. Sohncke. Halle, 1839. 

14. Marie, Maximilien. Histoire des Sciences Mathematiques et 

Physiques. Tome I.-XII. Paris, 1883-1888. 

15. Comte, A. Philosophy of Mathematics, translated by W. M. 


16. Hankel, Hermann. Die Entwickelung der Mathematik in den 

letzten Jahrhunderten. Tubingen, 1884. 

17. Gunther, Siegmund und Windelband, W. Geschichte der 

antiken Naturwissenschaft und Philosophic. Nordlingen, 1888. 

18. Arneth, A. Geschichte der reinen Mathematik. Stuttgart, 1852. 

19. Cantor, Moritz. Mathematische Beitrage zum Kulturleben der 

Volker. Halle, 1863. 

20. Matthiessen, Ludwig. Grundziige der Antiken und Modernen 

Algebra der Litteralen Gleichungen. Leipzig, 1878. 

21. Ohrtmann und Muller. Fortschritte der Mathematik. 

22. Peacock, George. Article "Arithmetic," in The Encyclopedia 

of Pure Mathematics. London, 1847. 

23. Herschel, J. F. W. Article "Mathematics," in Edinburgh 


24. Suter, Heinrich. Geschichte der Mathematischen Wissenschaf- 

ten. Zurich, 1873-75. 

25. QuETELET, A. Sciences Mathematiques et Physiques chez les 

Beiges. Bruxelles, 1866. 

26. PLAYFAIR, John. Article "Progress of the Mathematical and 

Physical Sciences," in Encyclopedia Britannica, 7th edition, 
continued in the 8th edition by Sir John Leslie. 

27. De Morgan, A. Arithmetical Books from the Invention of 

Printing to the Present Time. 

28. Napier, Mark. Memoirs of John Napier of Merchiston. 

Edinburgh, 1834. 

29. Halsted, G. B. "Note on the First English Euclid," American 

Journal of Mathematics, Vol. II., 1879. 


30. Madame Perier. The Life of Mr. Paschal. Translated into 

English by W. A., London, 1744. 

31. MONTUCLA, J. F. Histoire des Mathematiques. Paris, 1802. 

32. Duhring E. Kritische Geschichte der allgemeinen Principien der 

Mechanik. Leipzig, 1887. 

33. Brewster, D. The Memoirs of Newton. Edinburgh, 1860. 

34. Ball, W. W. R. A Short Account of the History of Mathematics. 

London, 1888, 2nd edition, 1893. 

35. De Morgan, A. "On the Early History of Infinitesimals," in the 

Philosophical Magazine, November, 1852. 

36. Bibliotheca Mathematica, herausgegeben von GuSTAF ENESTROM, 


37. Gunther, Siegmund. Vermischte Untersuchungen zur Geschich- 

te der mathematischen Wissenschaften. Leipzig, 1876. 

38. *Gerhardt, C. I. Geschichte der Mathematik in Deutschland. 

Miinchen, 1877. 

39. Gerhardt, C. I. Entdeckung der Differenzialrechnung durch 

Leibniz. Halle, 1848. 

40. Gerhardt, K. I. "Leibniz in London," in Sitzungsberichte der 

Koniglich Preussischen Academie der Wissenschaften zu Berlin, 
Februar, 1891. 

41. De Morgan, A. Articles "Fluxions" and "Commercium Epis- 

tolicum," in the Penny Cyclopedia. 

42. *TODHUNTER, I. A History of the Mathematical Theory of 

Probability from the Time of Pascal to that of Laplace. 
Cambridge and London, 1865. 

43. *TODHUNTER, I. A History of the Theory of Elasticity and of 

the Strength of Materials. Edited and completed by Karl 
Pearson. Cambridge, 1886. 

44. Todhunter, I. "Note on the History of Certain Formulae in 

Spherical Trigonometry," Philosophical Magazine, February, 

45. Die Basler Mathematiker, Daniel Bernoulli und Leonhard Euler. 

Basel, 1884. 


46. Reiff, R. Geschichte der Unendlichen Reihen. Tubingen, 1889. 

47. Waltershausen, W. Sartorius. Gauss, zum Gedachtniss. 

Leipzig, 1856. 

48. Baumgart, Oswald. Ueberdas Quadratische Reciprocitatsgesetz. 

Leipzig, 1885. 

49. Hathaway, A. S. "Early History of the Potential," Bulletin of 

the N. Y. Mathematical Society, I. 3. 

50. Wolf, Rudolf. Geschichte der Astronomic. Miinchen, 1877. 

51. Arago, D.F.J. "Eulogy on Laplace." Translated by B. Powell, 

Smithsonian Report, 1874. 

52. Beaumont, M. Elie De. "Memoir of Legendre." Translated by 

C. A. Alexander, Smithsonian Report, 1867. 

53. ARAGO, D. F. J. "Joseph Fourier." Smithsonian Report, 1871. 

54. Wiener, Christian. Lehrbuch der Darstellenden Geometric 

Leipzig, 1884. 

55. *LORlA, GlNO. Die Hauptsachlichsten Theorien der Geometrie 

in ihrer fruheren und heutigen Entwickelung, ins deutsche 
iibertragen von Fritz Schutte. Leipzig, 1888. 

56. Cayley, Arthur. Inaugural Address before the British Associa- 

tion, 1883. 

57. Spottiswoode, William. Inaugural Address before the British 

Association, 1878. 

58. Gibbs, J. Willard. "Multiple Algebra," Proceedings of the 

American Association for the Advancement of Science, 1886. 

59. Fink, Karl. Geschichte der Elementar-Mathematik. Tubingen, 


60. Wittstein, Armin. Zur Geschichte des Malfatti'schen Problems. 

Nordlingen, 1878. 

61. Klein, Felix. Vergleichende Betrachtungen iiber neuere geome- 

trische Forschungen. Erlangen, 1872. 

62. Forsyth, A. R. Theory of Functions of a Complex Variable. 

Cambridge, 1893. 

63. Graham, R. H. Geometry of Position. London, 1891. 


64. Schmidt, Franz. "Aus dem Leben zweier ungarischer Mathe- 

matiker Johann und Wolfgang Bolyai von Bolya." Grunert's 
Archiv, 48:2, 1868. 

65. Favaro, Anton. "Justus Bellavitis," Zeitschrift fiir Mathematik 

undPhysik, 26:5, 1881. 

66. Dronke, Ad. Julius Plucker. Bonn, 1871. 

67. Bauer, Gustav. Gedachtnissrede auf Otto Hesse. Miinchen, 


68. Alfred Clebsch. Versuch einer Darlegung und Wiirdigung 

seiner wissenschaftlichen Leistungen von einigen seiner Freunde. 
Leipzig, 1873. 

69. Haas, August. Versuch einer Darstellung der Geschichte des 

Krummungsmasses. Tubingen, 1881. 

70. Fine, Henry B. The Number- System of Algebra. Boston and 

New York, 1890. 

71. Schlegel, Victor. Hermann Grassmann, sein Leben und seine 

Werke. Leipzig, 1878. 

72. ZAHN, W. V. "Einige Worte zum Andenken an Hermann Hankel," 

Mathematische Annalen, VII. 4, 1874. 

73. MuiR, Thomas. A Treatise on Determinants. 1882. 

74. Salmon, George. "Arthur Cayley," Nature, 28:21, September, 


75. Cayley, A. "James Joseph Sylvester," Nature, 39:10, January, 


76. Burkhardt, Heinrich. "Die Anfange der Gruppentheorie 

und Paolo Ruffini," Zeitschrift fiir Mathematik und Physik, 
Supplement, 1892. 

77. Sylvester, J. J. Inaugural Presidential Address to the Mathemat- 

ical and Physical Section of the British Association at Exeter. 

78. Valson, C. A. La Vie et les travaux du Baron Cauchy. Tome I., 

II., Paris, 1868. 

79. Sachse, Arnold. Versuch einer Geschichte der Darstellung 

willkurlicher Funktionen einer variablen durch trigonometrische 
Reihen. Gottingen, 1879. 


80. Bois-Reymond, Pauldu. Zur Geschichte der Trigonometrischen 

Reihen, Eine Entgegnung. Tubingen. 

81. Poincare, Henri. Notice sur les Travaux Scientifiques de Henri 

Poincare. Paris, 1886. 

82. BjERKNES, C. A. Niels-Henrik Abel, Tableau de sa vie et de son 

action scientifique. Paris, 1885. 

83. Tucker, R. "Carl Friedrich Gauss," Nature, April, 1877. 

84. Dirichlet, Lejeune. Gedachtnissrede auf Carl Gustav Jacob 

Jacobi. 1852. 

85. Enneper, Alfred. Elliptische Funktionen. Theorie und Ge- 

schichte. Halle a/S., 1876. 

86. HENRICI, O. "Theory of Functions," Nature, 43:14 and 15, 1891. 

87. Darboux, Gaston. Notice sur les Travaux Scientifiques de M. 

Gaston Darboux. Paris, 1884. 

88. Kummer, E. E. Gedachtnissrede auf Gustav Peter Lejeune- Diri- 

chlet. Berlin, 1860. 

89. Smith, H. J. Stephen. "On the Present State and Prospects 

of Some Branches of Pure Mathematics," Proceedings of the 
London Mathematical Society, Vol. VIII, Nos. 104, 105, 1876. 

90. Glaisher, J. W. L. "Henry John Stephen Smith," Monthly 

Notices of the Royal Astronomical Society, XLIV., 4, 1884. 

91. Bessel als Bremer Handlungslehrling. Bremen, 1890. 

92. Frantz, J. Festrede aus Veranlassung von Bessel's hundertjahr- 

igem Geburtstag. Konigsberg, 1884. 

93. DziOBEK, O. Mathematical Theories of Planetary Motions. 

Translated into English by M. W. Harrington and W. J. Hussey. 

94. Hermite, Ch. "Discours prononce devant le president de 

la Republique," Bulletin des Sciences Mathematiques, XIV., 
Janvier, 1890. 

95. Schuster, Arthur. "The Influence of Mathematics on the 

Progress of Physics," Nature, 25:17, 1882. 

96. Kerbedz, E. de. "Sophie de Kowalevski," Rendiconti del Circolo 

Matematico di Palermo, V., 1891. 

97. VoiGT, W. Zum Gedachtniss von G. Kirchhoff. Gottingen, 1888. 


98. Bocher, Maxime. "A Bit of Mathematical History," Bulletin of 

the N. Y. Math. Soc, Vol. II., No. 5. 

99. Cayley, Arthur. Report on the Recent Progress of Theoretical 

Dynamics. 1857. 

100. Glazebrook, R. T. Report on Optical Theories. 1885. 

101. Rosenberger, F. Geschichte der Physik. Braunschweig, 1887- 





The contemplation of the various steps by which mankind 
has come into possession of the vast stock of mathematical 
knowledge can hardly fail to interest the mathematician. He 
takes pride in the fact that his science, more than any other, 
is an exact science, and that hardly anything ever done in 
mathematics has proved to be useless. The chemist smiles 
at the childish efforts of alchemists, but the mathematician 
finds the geometry of the Greeks and the arithmetic of the 
Hindoos as useful and admirable as any research of to-day. He 
is pleased to notice that though, in course of its development, 
mathematics has had periods of slow growth, yet in the main 
it has been pre-eminently a progressive science. 

The history of mathematics may be instructive as well 
as agreeable; it may not only remind us of what we have, 
but may also teach us how to increase our store. Says De 
Morgan, "The early history of the mind of men with regard 
to mathematics leads us to point out our own errors; and 
in this respect it is well to pay attention to the history of 
mathematics." It warns us against hasty conclusions; it points 
out the importance of a good notation upon the progress of 
the science; it discourages excessive specialisation on the part 
of investigators, by showing how apparently distinct branches 


have been found to possess unexpected connecting links; it 
saves the student from wasting time and energy upon problems 
which were, perhaps, solved long since; it discourages him 
from attacking an unsolved problem by the same method 
which has led other mathematicians to failure; it teaches that 
fortifications can be taken in other ways than by direct attack, 
that when repulsed from a direct assault it is well to reconnoitre 
and occupy the surrounding ground and to discover the secret 
paths by which the apparently unconquerable position can 
be taken. [1] The importance of this strategic rule may be 
emphasised by citing a case in which it has been violated. An 
untold amount of intellectual energy has been expended on 
the quadrature of the circle, yet no conquest has been made by 
direct assault. The circle-squarers have existed in crowds ever 
since the period of Archimedes. After innumerable failures 
to solve the problem at a time, even, when investigators 
possessed that most powerful tool, the differential calculus, 
persons versed in mathematics dropped the subject, while 
those who still persisted were completely ignorant of its 
history and generally misunderstood the conditions of the 
problem. "Our problem," says De Morgan, "is to square the 
circle with the old allowance of means: Euclid's postulates 
and nothing more. We cannot remember an instance in which 
a question to be solved by a definite method was tried by 
the best heads, and answered at last, by that method, after 
thousands of complete failures." But progress was made on 
this problem by approaching it from a different direction and 
by newly discovered paths. Lambert proved in 1761 that 


the ratio of the circumference of a circle to its diameter is 
incommensurable. Some years ago, Lindemann demonstrated 
that this ratio is also transcendental and that the quadrature 
of the circle, by means of the ruler and compass only, is 
impossible. He thus showed by actual proof that which keen- 
minded mathematicians had long suspected; namely, that the 
great army of circle-squarers have, for two thousand years, 
been assaulting a fortification which is as indestructible as the 
firmament of heaven. 

Another reason for the desirability of historical study is the 
value of historical knowledge to the teacher of mathematics. 
The interest which pupils take in their studies may be greatly 
increased if the solution of problems and the cold logic of 
geometrical demonstrations are interspersed with historical 
remarks and anecdotes. A class in arithmetic will be pleased 
to hear about the Hindoos and their invention of the "Arabic 
notation" ; they will marvel at the thousands of years which 
elapsed before people had even thought of introducing into 
the numeral notation that Columbus-egg — the zero; they 
will find it astounding that it should have taken so long to 
invent a notation which they themselves can now learn in a 
month. After the pupils have learned how to bisect a given 
angle, surprise them by telling of the many futile attempts 
which have been made to solve, by elementary geometry, 
the apparently very simple problem of the trisection of an 
angle. When they know how to construct a square whose 
area is double the area of a given square, tell them about the 
duplication of the cube — how the wrath of Apollo could be 


appeased only by the construction of a cubical altar double 
the given altar, and how mathematicians long wrestled with 
this problem. After the class have exhausted their energies 
on the theorem of the right triangle, tell them the legend 
about its discoverer — how Pythagoras, jubilant over his great 
accomplishment, sacrificed a hecatomb to the Muses who 
inspired him. When the value of mathematical training is 
called in question, quote the inscription over the entrance 
into the academy of Plato, the philosopher: "Let no one 
who is unacquainted with geometry enter here." Students 
in analytical geometry should know something of Descartes, 
and, after taking up the differential and integral calculus, they 
should become familiar with the parts that Newton, Leibniz, 
and Lagrange played in creating that science. In his historical 
talk it is possible for the teacher to make it plain to the student 
that mathematics is not a dead science, but a living one in 
which steady progress is made. [2] 

The history of mathematics is important also as a valuable 
contribution to the history of civilisation. Human progress 
is closely identified with scientific thought. Mathematical 
and physical researches are a reliable record of intellectual 
progress. The history of mathematics is one of the large 
windows through which the philosophic eye looks into past 
ages and traces the line of intellectual development. 



The fertile valley of the Euphrates and Tigris was one 
of the primeval seats of human society. Authentic history 
of the peoples inhabiting this region begins only with the 
foundation, in Chaldaea and Babylonia, of a united kingdom 
out of the previously disunited tribes. Much light has been 
thrown on their history by the discovery of the art of reading 
the cuneiform or wedge-shaped system of writing. 

In the study of Babylonian mathematics we begin with the 
notation of numbers. A vertical wedge \f stood for 1, while 
the characters < and V*- signified 10 and 100 respectively. 
Grotefend believes the character for 10 originally to have 
been the picture of two hands, as held in prayer, the palms 
being pressed together, the fingers close to each other, but the 
thumbs thrust out. In the Babylonian notation two principles 
were employed — the additive and multiplicative. Numbers 
below 100 were expressed by symbols whose respective values 
had to be added. Thus, If K stood for 2, K H for 3, ^iJ- for 4, 
<?» for 23, < << for 30. Here the symbols of higher order 
appear always to the left of those of lower order. In writing 
the hundreds, on the other hand, a smaller symbol was placed 
to the left of the 100, and was, in that case, to be multiplied 
by 100. Thus, < If *- signified 10 times 100, or 1000. But 


this symbol for 1000 was itself taken for a new unit, which 
could take smaller coefficients to its left. Thus, K. K. 1 *~ 
denoted, not 20 times 100, but 10 times 1000. Of the largest 
numbers written in cuneiform symbols, which have hitherto 
been found, none go as high as a million. [3] 

If, as is believed by most specialists, the early Sumerians 
were the inventors of the cuneiform writing, then they were, 
in all probability, also familiar with the notation of numbers. 
Most surprising, in this connection, is the fact that Sumerian 
inscriptions disclose the use, not only of the above decimal 
system, but also of a sexagesimal one. The latter was used 
chiefly in constructing tables for weights and measures. It 
is full of historical interest. Its consequential development, 
both for integers and fractions, reveals a high degree of 
mathematical insight. We possess two Babylonian tablets 
which exhibit its use. One of them, probably written between 
2300 and 1600 B.C., contains a table of square numbers up 
to 60 2 . The numbers 1, 4, 9, 16, 25, 36, 49, are given as the 
squares of the first seven integers respectively. We have next 
1.4 = 8 2 , 1.21 = 9 2 , 1.40 = 10 2 , 2.1 = ll 2 , etc. This remains 
unintelligible, unless we assume the sexagesimal scale, which 
makes 1.4 = 60 + 4, 1.21 = 60 + 21, 2.1 = 2.60 + 1. The second 
tablet records the magnitude of the illuminated portion of the 
moon's disc for every day from new to full moon, the whole 
disc being assumed to consist of 240 parts. The illuminated 
parts during the first five days are the series 5, 10, 20, 40, 
1.20(= 80), which is a geometrical progression. From here on 
the series becomes an arithmetical progression, the numbers 


from the fifth to the fifteenth day being respectively 1.20, 1.36, 
1.52, 1.8, 2.24, 2.40, 2.56, 3.12, 3.28, 3.44, 4. This table not only 
exhibits the use of the sexagesimal system, but also indicates 
the acquaintance of the Babylonians with progressions. Not 
to be overlooked is the fact that in the sexagesimal notation 
of integers the "principle of position" was employed. Thus, 
in 1.4 (= 64), the 1 is made to stand for 60, the unit of 
the second order, by virtue of its position with respect to 
the 4. The introduction of this principle at so early a date 
is the more remarkable, because in the decimal notation 
it was not introduced till about the fifth or sixth century 
after Christ. The principle of position, in its general and 
systematic application, requires a symbol for zero. We ask, 
Did the Babylonians possess one? Had they already taken the 
gigantic step of representing by a symbol the absence of units? 
Neither of the above tables answers this question, for they 
happen to contain no number in which there was occasion to 
use a zero. The sexagesimal system was used also in fractions. 
Thus, in the Babylonian inscriptions, \ and \ are designated 
by 30 and 20, the reader being expected, in his mind, to 
supply the word "sixtieths." The Greek geometer Hypsicles 
and the Alexandrian astronomer Ptolemaeus borrowed the 
sexagesimal notation of fractions from the Babylonians and 
introduced it into Greece. From that time sexagesimal frac- 
tions held almost full sway in astronomical and mathematical 
calculations until the sixteenth century, when they finally 
yielded their place to the decimal fractions. It may be asked, 
What led to the invention of the sexagesimal system? Why 


was it that 60 parts were selected? To this we have no positive 
answer. Ten was chosen, in the decimal system, because it 
represents the number of fingers. But nothing of the human 
body could have suggested 60. Cantor offers the following 
theory: At first the Babylonians reckoned the year at 360 days. 
This led to the division of the circle into 360 degrees, each 
degree representing the daily amount of the supposed yearly 
revolution of the sun around the earth. Now they were, very 
probably, familiar with the fact that the radius can be applied 
to its circumference as a chord 6 times, and that each of 
these chords subtends an arc measuring exactly 60 degrees. 
Fixing their attention upon these degrees, the division into 
60 parts may have suggested itself to them. Thus, when 
greater precision necessitated a subdivision of the degree, it 
was partitioned into 60 minutes. In this way the sexagesimal 
notation may have originated. The division of the day into 
24 hours, and of the hour into minutes and seconds on the 
scale of 60, is due to the Babylonians. 

It appears that the people in the Tigro-Euphrates basin had 
made very creditable advance in arithmetic. Their knowledge 
of arithmetical and geometrical progressions has already been 
alluded to. Iamblichus attributes to them also a knowledge 
of proportion, and even the invention of the so-called musical 
proportion. Though we possess no conclusive proof, we have 
nevertheless reason to believe that in practical calculation 
they used the abacus. Among the races of middle Asia, 
even as far as China, the abacus is as old as fable. Now, 
Babylon was once a great commercial centre, — the metropolis 


of many nations, — and it is, therefore, not unreasonable to 
suppose that her merchants employed this most improved aid 
to calculation. 

In geometry the Babylonians accomplished almost nothing. 
Besides the division of the circumference into 6 parts by its 
radius, and into 360 degrees, they had some knowledge of 
geometrical figures, such as the triangle and quadrangle, which 
they used in their auguries. Like the Hebrews (1 Kin. 7:23), 
they took it = 3. Of geometrical demonstrations there is, of 
course, no trace. "As a rule, in the Oriental mind the intuitive 
powers eclipse the severely rational and logical." 

The astronomy of the Babylonians has attracted much 
attention. They worshipped the heavenly bodies from the 
earliest historic times. When Alexander the Great, after 
the battle of Arbela (331 B.C.), took possession of Babylon, 
Callisthenes found there on burned brick astronomical records 
reaching back as far as 2234 B.C. Porphyrius says that 
these were sent to Aristotle. Ptolemy, the Alexandrian 
astronomer, possessed a Babylonian record of eclipses going 
back to 747 B.C. Recently Epping and Strassmaier [4] threw 
considerable light on Babylonian chronology and astronomy 
by explaining two calendars of the years 123 B.C. and 111 B.C., 
taken from cuneiform tablets coming, presumably, from an 
old observatory. These scholars have succeeded in giving an 
account of the Babylonian calculation of the new and full 
moon, and have identified by calculations the Babylonian 
names of the planets, and of the twelve zodiacal signs and 
twenty-eight normal stars which correspond to some extent 


with the twenty-eight nakshatras of the Hindoos. We append 
part of an Assyrian astronomical report, as translated by 

"To the King, my lord, thy faithful servant, Mar-Istar." 

"... On the first day, as the new moon's day of the month Thammuz 
declined, the moon was again visible over the planet Mercury, as I had 
already predicted to my master the King. I erred not." 


Though there is great difference of opinion regarding the 
antiquity of Egyptian civilisation, yet all authorities agree in 
the statement that, however far back they go, they find no 
uncivilised state of society. "Menes, the first king, changes 
the course of the Nile, makes a great reservoir, and builds 
the temple of Phthah at Memphis." The Egyptians built the 
pyramids at a very early period. Surely a people engaging in 
enterprises of such magnitude must have known something of 
mathematics — at least of practical mathematics. 

All Greek writers are unanimous in ascribing, without 
envy, to Egypt the priority of invention in the mathematical 
sciences. Plato in Phcedrus says: "At the Egyptian city 
of Naucratis there was a famous old god whose name was 
Theuth; the bird which is called the Ibis was sacred to him, 
and he was the inventor of many arts, such as arithmetic and 
calculation and geometry and astronomy and draughts and 
dice, but his great discovery was the use of letters." 

Aristotle says that mathematics had its birth in Egypt, 
because there the priestly class had the leisure needful for the 


study of it. Geometry, in particular, is said by Herodotus, 
Diodorus, Diogenes Laertius, Iamblichus, and other ancient 
writers to have originated in Egypt. [5] In Herodotus we find 
this (II. c. 109): "They said also that this king [Sesostris] 
divided the land among all Egyptians so as to give each one a 
quadrangle of equal size and to draw from each his revenues, 
by imposing a tax to be levied yearly. But every one from 
whose part the river tore away anything, had to go to him 
and notify what had happened; he then sent the overseers, 
who had to measure out by how much the land had become 
smaller, in order that the owner might pay on what was left, in 
proportion to the entire tax imposed. In this way, it appears 
to me, geometry originated, which passed thence to Hellas." 

We abstain from introducing additional Greek opinion 
regarding Egyptian mathematics, or from indulging in wild 
conjectures. We rest our account on documentary evidence. 
A hieratic papyrus, included in the Rhind collection of the 
British Museum, was deciphered by Eisenlohr in 1877, and 
found to be a mathematical manual containing problems in 
arithmetic and geometry. It was written by Ahmes some 
time before 1700 B.C., and was founded on an older work 
believed by Birch to date back as far as 3400 B.C.! This curious 
papyrus — the most ancient mathematical handbook known 
to us — puts us at once in contact with the mathematical 
thought in Egypt of three or five thousand years ago. It is 
entitled "Directions for obtaining the Knowledge of all Dark 
Things." We see from it that the Egyptians cared but little 
for theoretical results. Theorems are not found in it at all. It 


contains "hardly any general rules of procedure, but chiefly 
mere statements of results intended possibly to be explained 
by a teacher to his pupils." [6] In geometry the forte of 
the Egyptians lay in making constructions and determining 
areas. The area of an isosceles triangle, of which the sides 
measure 10 ruths and the base 4 ruths, was erroneously given 
as 20 square ruths, or half the product of the base by one 
side. The area of an isosceles trapezoid is found, similarly, 
by multiplying half the sum of the parallel sides by one of 
the non-parallel sides. The area of a circle is found by 
deducting from the diameter g of its length and squaring the 
remainder. Here n is taken = (t^) 2 = 3.1604. . ., a very fair 
approximation. [6] The papyrus explains also such problems 
as these, — To mark out in the field a right triangle whose sides 
are 10 and 4 units; or a trapezoid whose parallel sides are 6 
and 4, and the non-parallel sides each 20 units. 

Some problems in this papyrus seem to imply a rudimentary 
knowledge of proportion. 

The base-lines of the pyramids run north and south, and 
east and west, but probably only the lines running north and 
south were determined by astronomical observations. This, 
coupled with the fact that the word harpedonaptds, applied to 
Egyptian geometers, means "rope-stretchers," would point to 
the conclusion that the Egyptian, like the Indian and Chinese 
geometers, constructed a right triangle upon a given line, by 
stretching around three pegs a rope consisting of three parts 
in the ratios 3:4:5, and thus forming a right triangle. [3] If 
this explanation is correct, then the Egyptians were familiar, 


2000 years B.C., with the well-known property of the right 
triangle, for the special case at least when the sides are in the 
ratio 3:4:5. 

On the walls of the celebrated temple of Horus at Edfu 
have been found hieroglyphics, written about 100 B.C., which 
enumerate the pieces of land owned by the priesthood, and 
give their areas. The area of any quadrilateral, however 

irregular, is there found by the formula . Thus, 

for a quadrangle whose opposite sides are 5 and 8, 20 and 15, 
is given the area 113i i. [7] The incorrect formulae of Ahmes 
of 3000 years B.C. yield generally closer approximations than 
those of the Edfu inscriptions, written 200 years after Euclid! 

The fact that the geometry of the Egyptians consists chiefly 
of constructions, goes far to explain certain of its great defects. 
The Egyptians failed in two essential points without which 
a science of geometry, in the true sense of the word, cannot 
exist. In the first place, they failed to construct a rigorously 
logical system of geometry, resting upon a few axioms and 
postulates. A great many of their rules, especially those in 
solid geometry, had probably not been proved at all, but were 
known to be true merely from observation or as matters of 
fact. The second great defect was their inability to bring the 
numerous special cases under a more general view, and thereby 
to arrive at broader and more fundamental theorems. Some of 
the simplest geometrical truths were divided into numberless 
special cases of which each was supposed to require separate 

Some particulars about Egyptian geometry can be men- 


tioned more advantageously in connection with the early 
Greek mathematicians who came to the Egyptian priests for 

An insight into Egyptian methods of numeration was ob- 
tained through the ingenious deciphering of the hieroglyphics 
by Champollion, Young, and their successors. The symbols 
used were the following: 1 for 1, H for 10, 0? for 100, 
I for 1000, f for 10,000, ^ for 100,000, )£ for 1,000,000, _Q_ 
for 10, 000, 000. [3] The symbol for 1 represents a vertical staff; 
that for 10, 000 a pointing finger; that for 100, 000 a burbot; 
that for 1,000,000, a man in astonishment. The significance 
of the remaining symbols is very doubtful. The writing of 
numbers with these hieroglyphics was very cumbrous. The 
unit symbol of each order was repeated as many times as there 
were units in that order. The principle employed was the 
additive. Thus, 23 was written ffi ffi 1 1 1 . 

Besides the hieroglyphics, Egypt possesses the hieratic and 
demotic writings, but for want of space we pass them by. 

Herodotus makes an important statement concerning the 
mode of computing among the Egyptians. He says that they 
" calculate with pebbles by moving the hand from right to 
left, while the Hellenes move it from left to right." Herein 
we recognise again that instrumental method of figuring so 
extensively used by peoples of antiquity. The Egyptians used 
the decimal scale. Since, in figuring, they moved their hands 
horizontally, it seems probable that they used ciphering- 
boards with vertical columns. In each column there must 
have been not more than nine pebbles, for ten pebbles would 


be equal to one pebble in the column next to the left. 

The Ahmes papyrus contains interesting information on 

the way in which the Egyptians employed fractions. Their 

methods of operation were, of course, radically different from 

ours. Fractions were a subject of very great difficulty with 

the ancients. Simultaneous changes in both numerator and 

denominator were usually avoided. In manipulating fractions 

the Babylonians kept the denominators (60) constant. The 

Romans likewise kept them constant, but equal to 12. The 

Egyptians and Greeks, on the other hand, kept the numerators 

constant, and dealt with variable denominators. Ahmes used 

the term "fraction" in a restricted sense, for he applied 

it only to unit- fractions, or fractions having unity for the 

numerator. It was designated by writing the denominator 

and then placing over it a dot. Fractional values which could 

not be expressed by any one unit-fraction were expressed as 

the sum of two or more of them. Thus, he wrote | j§ in place 

of |. The first important problem naturally arising was, how 

to represent any fractional value as the sum of unit-fractions. 

This was solved by aid of a table, given in the papyrus, in 


which all fractions of the form (where n designates 

2n + 1 v 6 

successively all the numbers up to 49) are reduced to the sum 
of unit-fractions. Thus, 7 = 5^; w = M TM- When, by 
whom, and how this table was calculated, we do not know. 
Probably it was compiled empirically at different times, by 
different persons. It will be seen that by repeated application 
of this table, a fraction whose numerator exceeds two can be 
expressed in the desired form, provided that there is a fraction 


in the table having the same denominator that it has. Take, 
for example, the problem, to divide 5 by 21. In the first place, 
5 = 1 + 2 + 2. From the table we get ^ = ^ ^. Then ^ = 
J_^rJ_J_^/'J_J_ N i - i i /ll^ _ J_IJ_ _ 11 _ 1J_J_ 

21 ~*~ 1 14 42 ) t 1 14 42 J ~ 21 "+" U4 42 J ~ 21 7 21 ~ 7 21 ~ 7 14 42 • 

The papyrus contains problems in which it is required that 
fractions be raised by addition or multiplication to given whole 
numbers or to other fractions. For example, it is required to 
increase | g iq jq 43 to 1. The common denominator taken 
appears to be 45, for the numbers are stated as 11 1, 5^ g, 4^, 
1^, 1. The sum of these is 23^ \ \ forty-fifths. Add to this 
\ 4^, and the sum is \. Add |, and we have 1. Hence the 
quantity to be added to the given fraction is \ g ^j. 

Having finished the subject of fractions, Ahmes proceeds 
to the solution of equations of one unknown quantity. The 
unknown quantity is called 'hau' or heap. Thus the problem, 
"heap, its ^, its whole, it makes 19," i.e. — + x = 19. In this 

case, the solution is as follows: — = 19; — = 2| |; x = 16^ g- 
But in other problems, the solutions are effected by various 
other methods. It thus appears that the beginnings of algebra 
are as ancient as those of geometry. 

The principal defect of Egyptian arithmetic was the lack of 
a simple, comprehensive symbolism — a defect which not even 
the Greeks were able to remove. 

The Ahmes papyrus doubtless represents the most advanced 
attainments of the Egyptians in arithmetic and geometry. It is 
remarkable that they should have reached so great proficiency 
in mathematics at so remote a period of antiquity. But 


strange, indeed, is the fact that, during the next two thousand 
years, they should have made no progress whatsoever in 
it. The conclusion forces itself upon us, that they resemble 
the Chinese in the stationary character, not only of their 
government, but also of their learning. All the knowledge of 
geometry which they possessed when Greek scholars visited 
them, six centuries B.C., was doubtless known to them two 
thousand years earlier, when they built those stupendous 
and gigantic structures — the pyramids. An explanation for 
this stagnation of learning has been sought in the fact that 
their early discoveries in mathematics and medicine had the 
misfortune of being entered upon their sacred books and that, 
in after ages, it was considered heretical to augment or modify 
anything therein. Thus the books themselves closed the gates 
to progress. 



About the seventh century B.C. an active commercial in- 
tercourse sprang up between Greece and Egypt. Naturally 
there arose an interchange of ideas as well as of merchan- 
dise. Greeks, thirsting for knowledge, sought the Egyptian 
priests for instruction. Thales, Pythagoras, (Enopides, Plato, 
Democritus, Eudoxus, all visited the land of the pyramids. 
Egyptian ideas were thus transplanted across the sea and there 
stimulated Greek thought, directed it into new lines, and gave 
to it a basis to work upon. Greek culture, therefore, is not 


primitive. Not only in mathematics, but also in mythology 
and art, Hellas owes a debt to older countries. To Egypt 
Greece is indebted, among other things, for its elementary 
geometry. But this does not lessen our admiration for the 
Greek mind. From the moment that Hellenic philosophers 
applied themselves to the study of Egyptian geometry, this 
science assumed a radically different aspect. "Whatever we 
Greeks receive, we improve and perfect," says Plato. The 
Egyptians carried geometry no further than was absolutely 
necessary for their practical wants. The Greeks, on the other 
hand, had within them a strong speculative tendency. They 
felt a craving to discover the reasons for things. They found 
pleasure in the contemplation of ideal relations, and loved 
science as science. 

Our sources of information on the history of Greek geometry 
before Euclid consist merely of scattered notices in ancient 
writers. The early mathematicians, Thales and Pythagoras, 
left behind no written records of their discoveries. A full 
history of Greek geometry and astronomy during this period, 
written by Eudemus, a pupil of Aristotle, has been lost. It was 
well known to Proclus, who, in his commentaries on Euclid, 
gives a brief account of it. This abstract constitutes our most 
reliable information. We shall quote it frequently under the 
name of Eudemian Summary. 


The Ionic School. 

To Thales of Miletus (640-546 B.C.), one of the "seven wise 
men," and the founder of the Ionic school, falls the honour 
of having introduced the study of geometry into Greece. 
During middle life he engaged in commercial pursuits, which 
took him to Egypt. He is said to have resided there, and 
to have studied the physical sciences and mathematics with 
the Egyptian priests. Plutarch declares that Thales soon 
excelled his masters, and amazed King Amasis by measuring 
the heights of the pyramids from their shadows. According to 
Plutarch, this was done by considering that the shadow cast 
by a vertical staff of known length bears the same ratio to the 
shadow of the pyramid as the height of the staff bears to the 
height of the pyramid. This solution presupposes a knowledge 
of proportion, and the Ahmes papyrus actually shows that 
the rudiments of proportion were known to the Egyptians. 
According to Diogenes Laertius, the pyramids were measured 
by Thales in a different way; viz. by finding the length of the 
shadow of the pyramid at the moment when the shadow of a 
staff was equal to its own length. 

The Eudemian Summary ascribes to Thales the invention 
of the theorems on the equality of vertical angles, the equality 
of the angles at the base of an isosceles triangle, the bisection 
of a circle by any diameter, and the congruence of two 
triangles having a side and the two adjacent angles equal 
respectively. The last theorem he applied to the measurement 
of the distances of ships from the shore. Thus Thales was 


the first to apply theoretical geometry to practical uses. The 
theorem that all angles inscribed in a semicircle are right 
angles is attributed by some ancient writers to Thales, by 
others to Pythagoras. Thales was doubtless familiar with 
other theorems, not recorded by the ancients. It has been 
inferred that he knew the sum of the three angles of a triangle 
to be equal to two right angles, and the sides of equiangular 
triangles to be proportional. [8] The Egyptians must have 
made use of the above theorems on the straight line, in some of 
their constructions found in the Ahmes papyrus, but it was left 
for the Greek philosopher to give these truths, which others 
saw, but did not formulate into words, an explicit, abstract 
expression, and to put into scientific language and subject 
to proof that which others merely felt to be true. Thales 
may be said to have created the geometry of lines, essentially 
abstract in its character, while the Egyptians studied only 
the geometry of surfaces and the rudiments of solid geometry, 
empirical in their character. [8] 

With Thales begins also the study of scientific astronomy. 
He acquired great celebrity by the prediction of a solar eclipse 
in 585 B.C. Whether he predicted the day of the occurrence, 
or simply the year, is not known. It is told of him that while 
contemplating the stars during an evening walk, he fell into a 
ditch. The good old woman attending him exclaimed, "How 
canst thou know what is doing in the heavens, when thou seest 
not what is at thy feet?" 

The two most prominent pupils of Thales were Anaxi- 
mander (b. 611 B.C.) and Anaximenes (b. 570 B.C.). They 


studied chiefly astronomy and physical philosophy. Of Anax- 
agoras, a pupil of Anaximenes, and the last philosopher of 
the Ionic school, we know little, except that, while in prison, 
he passed his time attempting to square the circle. This is 
the first time, in the history of mathematics, that we find 
mention of the famous problem of the quadrature of the 
circle, that rock upon which so many reputations have been 
destroyed. It turns upon the determination of the exact value 
of ir. Approximations to n had been made by the Chinese, 
Babylonians, Hebrews, and Egyptians. But the invention of 
a method to find its exact value, is the knotty problem which 
has engaged the attention of many minds from the time of 
Anaxagoras down to our own. Anaxagoras did not offer any 
solution of it, and seems to have luckily escaped paralogisms. 

About the time of Anaxagoras, but isolated from the Ionic 
school, flourished (Enopides of Chios. Proclus ascribes to 
him the solution of the following problems: From a point 
without, to draw a perpendicular to a given line, and to 
draw an angle on a line equal to a given angle. That a man 
could gain a reputation by solving problems so elementary 
as these, indicates that geometry was still in its infancy, and 
that the Greeks had not yet gotten far beyond the Egyptian 

The Ionic school lasted over one hundred years. The 
progress of mathematics during that period was slow, as 
compared with its growth in a later epoch of Greek history. A 
new impetus to its progress was given by Pythagoras. 


The School of Pythagoras. 

Pythagoras (5807-500? B.C.) was one of those figures 
which impressed the imagination of succeeding times to such 
an extent that their real histories have become difficult to be 
discerned through the mythical haze that envelops them. The 
following account of Pythagoras excludes the most doubtful 
statements. He was a native of Samos, and was drawn by the 
fame of Pherecydes to the island of Syros. He then visited 
the ancient Thales, who incited him to study in Egypt. He 
sojourned in Egypt many years, and may have visited Babylon. 
On his return to Samos, he found it under the tyranny of 
Polycrates. Failing in an attempt to found a school there, he 
quitted home again and, following the current of civilisation, 
removed to Magna Graecia in South Italy. He settled at 
Croton, and founded the famous Pythagorean school. This 
was not merely an academy for the teaching of philosophy, 
mathematics, and natural science, but it was a brotherhood, 
the members of which were united for life. This brotherhood 
had observances approaching masonic peculiarity. They were 
forbidden to divulge the discoveries and doctrines of their 
school. Hence we are obliged to speak of the Pythagoreans 
as a body, and find it difficult to determine to whom each 
particular discovery is to be ascribed. The Pythagoreans 
themselves were in the habit of referring every discovery back 
to the great founder of the sect. 

This school grew rapidly and gained considerable political 
ascendency. But the mystic and secret observances, intro- 


duced in imitation of Egyptian usages, and the aristocratic 
tendencies of the school, caused it to become an object of 
suspicion. The democratic party in Lower Italy revolted and 
destroyed the buildings of the Pythagorean school. Pythag- 
oras fled to Tarentum and thence to Metapontum, where he 
was murdered. 

Pythagoras has left behind no mathematical treatises, and 
our sources of information are rather scanty. Certain it is that, 
in the Pythagorean school, mathematics was the principal 
study. Pythagoras raised mathematics to the rank of a science. 
Arithmetic was courted by him as fervently as geometry. In 
fact, arithmetic is the foundation of his philosophic system. 

The Eudemian Summary says that "Pythagoras changed 
the study of geometry into the form of a liberal education, 
for he examined its principles to the bottom, and investigated 
its theorems in an immaterial and intellectual manner." His 
geometry was connected closely with his arithmetic. He was 
especially fond of those geometrical relations which admitted 
of arithmetical expression. 

Like Egyptian geometry, the geometry of the Pythagoreans 
is much concerned with areas. To Pythagoras is ascribed the 
important theorem that the square on the hypotenuse of a 
right triangle is equal to the sum of the squares on the other 
two sides. He had probably learned from the Egyptians the 
truth of the theorem in the special case when the sides are 
3, 4, 5, respectively. The story goes, that Pythagoras was so 
jubilant over this discovery that he sacrificed a hecatomb. Its 
authenticity is doubted, because the Pythagoreans believed 


in the transmigration of the soul and opposed, therefore, 
the shedding of blood. In the later traditions of the Neo- 
Pythagoreans this objection is removed by replacing this 
bloody sacrifice by that of "an ox made of flour" ! The proof 
of the law of three squares, given in Euclid's Elements, I. 47, 
is due to Euclid himself, and not to the Pythagoreans. What 
the Pythagorean method of proof was has been a favourite 
topic for conjecture. 

The theorem on the sum of the three angles of a triangle, 
presumably known to Thales, was proved by the Pythagoreans 
after the manner of Euclid. They demonstrated also that 
the plane about a point is completely filled by six equilateral 
triangles, four squares, or three regular hexagons, so that it is 
possible to divide up a plane into figures of either kind. 

From the equilateral triangle and the square arise the 
solids, namely the tetraedron, octaedron, icosaedron, and 
the cube. These solids were, in all probability, known 
to the Egyptians, excepting, perhaps, the icosaedron. In 
Pythagorean philosophy, they represent respectively the four 
elements of the physical world; namely, fire, air, water, and 
earth. Later another regular solid was discovered, namely the 
dodecaedron, which, in absence of a fifth element, was made to 
represent the universe itself. Iamblichus states that Hippasus, 
a Pythagorean, perished in the sea, because he boasted that 
he first divulged "the sphere with the twelve pentagons." The 
star-shaped pentagram was used as a symbol of recognition 
by the Pythagoreans, and was called by them Health. 

Pythagoras called the sphere the most beautiful of all 


solids, and the circle the most beautiful of all plane figures. 
The treatment of the subjects of proportion and of irrational 
quantities by him and his school will be taken up under the 
head of arithmetic. 

According to Eudemus, the Pythagoreans invented the 
problems concerning the application of areas, including the 
cases of defect and excess, as in Euclid, VI. 28, 29. 

They were also familiar with the construction of a polygon 
equal in area to a given polygon and similar to another given 
polygon. This problem depends upon several important and 
somewhat advanced theorems, and testifies to the fact that 
the Pythagoreans made no mean progress in geometry. 

Of the theorems generally ascribed to the Italian school, 
some cannot be attributed to Pythagoras himself, nor to his 
earliest successors. The progress from empirical to reasoned 
solutions must, of necessity, have been slow. It is worth 
noticing that on the circle no theorem of any importance was 
discovered by this school. 

Though politics broke up the Pythagorean fraternity, yet 
the school continued to exist at least two centuries longer. 
Among the later Pythagoreans, Philolaus and Archytas are the 
most prominent. Philolaus wrote a book on the Pythagorean 
doctrines. By him were first given to the world the teachings 
of the Italian school, which had been kept secret for a whole 
century. The brilliant Archytas of Tarentum (428-347 B.C.), 
known as a great statesman and general, and universally 
admired for his virtues, was the only great geometer among 
the Greeks when Plato opened his school. Archytas was 


the first to apply geometry to mechanics and to treat the 
latter subject methodically. He also found a very ingenious 
mechanical solution to the problem of the duplication of the 
cube. His solution involves clear notions on the generation of 
cones and cylinders. This problem reduces itself to finding 
two mean proportionals between two given lines. These mean 
proportionals were obtained by Archytas from the section of 
a half-cylinder. The doctrine of proportion was advanced 
through him. 

There is every reason to believe that the later Pythagoreans 
exercised a strong influence on the study and development of 
mathematics at Athens. The Sophists acquired geometry from 
Pythagorean sources. Plato bought the works of Philolaus, 
and had a warm friend in Archytas. 

The Sophist School. 

After the defeat of the Persians under Xerxes at the battle 
of Salamis, 480 B.C., a league was formed among the Greeks 
to preserve the freedom of the now liberated Greek cities on 
the islands and coast of the iEgaean Sea. Of this league Athens 
soon became leader and dictator. She caused the separate 
treasury of the league to be merged into that of Athens, and 
then spent the money of her allies for her own aggrandisement. 
Athens was also a great commercial centre. Thus she became 
the richest and most beautiful city of antiquity. All menial 
work was performed by slaves. The citizen of Athens was well- 
to-do and enjoyed a large amount of leisure. The government 


being purely democratic, every citizen was a politician. To 
make his influence felt among his fellow-men he must, first 
of all, be educated. Thus there arose a demand for teachers. 
The supply came principally from Sicily, where Pythagorean 
doctrines had spread. These teachers were called Sophists, or 
"wise men." Unlike the Pythagoreans, they accepted pay for 
their teaching. Although rhetoric was the principal feature 
of their instruction, they also taught geometry, astronomy, 
and philosophy. Athens soon became the headquarters of 
Grecian men of letters, and of mathematicians in particular. 
The home of mathematics among the Greeks was first in the 
Ionian Islands, then in Lower Italy, and during the time now 
under consideration, at Athens. 

The geometry of the circle, which had been entirely ne- 
glected by the Pythagoreans, was taken up by the Sophists. 
Nearly all their discoveries were made in connection with their 
innumerable attempts to solve the following three famous 

(1) To trisect an arc or an angle. 

(2) To "double the cube," i.e. to find a cube whose volume 
is double that of a given cube. 

(3) To "square the circle," i. e. to find a square or some other 
rectilinear figure exactly equal in area to a given circle. 

These problems have probably been the subject of more 
discussion and research than any other problems in math- 
ematics. The bisection of an angle was one of the easiest 
problems in geometry. The trisection of an angle, on the other 


hand, presented unexpected difficulties. A right angle had 
been divided into three equal parts by the Pythagoreans. But 
the general problem, though easy in appearance, transcended 
the power of elementary geometry. Among the first to wrestle 
with it was Hippias of Elis, a contemporary of Socrates, and 
born about 460 B.C. Like all the later geometers, he failed 
in effecting the trisection by means of a ruler and compass 
only. Proclus mentions a man, Hippias, presumably Hippias 
of Elis, as the inventor of a transcendental curve which served 
to divide an angle not only into three, but into any number of 
equal parts. This same curve was used later by Deinostratus 
and others for the quadrature of the circle. On this account it 
is called the quadratrix. 

The Pythagoreans had shown that the diagonal of a square 
is the side of another square having double the area of the 
original one. This probably suggested the problem of the 
duplication of the cube, i. e. to find the edge of a cube having 
double the volume of a given cube. Eratosthenes ascribes 
to this problem a different origin. The Delians were once 
suffering from a pestilence and were ordered by the oracle 
to double a certain cubical altar. Thoughtless workmen 
simply constructed a cube with edges twice as long, but this 
did not pacify the gods. The error being discovered, Plato 
was consulted on the matter. He and his disciples searched 
eagerly for a solution to this "Delian Problem." Hippocrates 
of Chios (about 430 B.C.), a talented mathematician, but 
otherwise slow and stupid, was the first to show that the 
problem could be reduced to finding two mean proportionals 


between a given line and another twice as long. For, in the 
proportion a : x = x : y = y : 2a, since x 1 = ay and y 2 = 2ax 
and a; 4 = a 2 y 2 , we have x 4 = 2c? x and x 3 = 2a 3 . But he failed 
to find the two mean proportionals. His attempt to square the 
circle was also a failure; for though he made himself celebrated 
by squaring a lune, he committed an error in attempting to 
apply this result to the squaring of the circle. 

In his study of the quadrature and duplication-problems, 
Hippocrates contributed much to the geometry of the circle. 

The subject of similar figures was studied and partly 
developed by Hippocrates. This involved the theory of 
proportion. Proportion had, thus far, been used by the 
Greeks only in numbers. They never succeeded in uniting the 
notions of numbers and magnitudes. The term "number" was 
used by them in a restricted sense. What we call irrational 
numbers was not included under this notion. Not even rational 
fractions were called numbers. They used the word in the same 
sense as we use "integers." Hence numbers were conceived 
as discontinuous, while magnitudes were continuous. The 
two notions appeared, therefore, entirely distinct. The chasm 
between them is exposed to full view in the statement of 
Euclid that "incommensurable magnitudes do not have the 
same ratio as numbers." In Euclid's Elements we find the 
theory of proportion of magnitudes developed and treated 
independent of that of numbers. The transfer of the theory 
of proportion from numbers to magnitudes (and to lengths in 
particular) was a difficult and important step. 

Hippocrates added to his fame by writing a geometrical 


text-book, called the Elements. This publication shows 
that the Pythagorean habit of secrecy was being abandoned; 
secrecy was contrary to the spirit of Athenian life. 

The Sophist Antiphon, a contemporary of Hippocrates, 
introduced the process of exhaustion for the purpose of 
solving the problem of the quadrature. He did himself credit 
by remarking that by inscribing in a circle a square, and on 
its sides erecting isosceles triangles with their vertices in the 
circumference, and on the sides of these triangles erecting 
new triangles, etc., one could obtain a succession of regular 
polygons of 8, 16, 32, 64 sides, and so on, of which each 
approaches nearer to the circle than the previous one, until 
the circle is finally exhausted. Thus is obtained an inscribed 
polygon whose sides coincide with the circumference. Since 
there can be found squares equal in area to any polygon, there 
also can be found a square equal to the last polygon inscribed, 
and therefore equal to the circle itself. Bryson of Heraclea, 
a contemporary of Antiphon, advanced the problem of the 
quadrature considerably by circumscribing polygons at the 
same time that he inscribed polygons. He erred, however, 
in assuming that the area of a circle was the arithmetical 
mean between circumscribed and inscribed polygons. Unlike 
Bryson and the rest of Greek geometers, Antiphon seems to 
have believed it possible, by continually doubling the sides of 
an inscribed polygon, to obtain a polygon coinciding with the 
circle. This question gave rise to lively disputes in Athens. If 
a polygon can coincide with the circle, then, says Simplicius, 
we must put aside the notion that magnitudes are divisible 


ad infinitum. Aristotle always supported the theory of the 
infinite divisibility, while Zeno, the Stoic, attempted to show 
its absurdity by proving that if magnitudes are infinitely 
divisible, motion is impossible. Zeno argues that Achilles 
could not overtake a tortoise; for while he hastened to the 
place where the tortoise had been when he started, the tortoise 
crept some distance ahead, and while Achilles reached that 
second spot, the tortoise again moved forward a little, and so 
on. Thus the tortoise was always in advance of Achilles. Such 
arguments greatly confounded Greek geometers. No wonder 
they were deterred by such paradoxes from introducing the 
idea of infinity into their geometry. It did not suit the rigour 
of their proofs. 

The process of Antiphon and Bryson gave rise to the cum- 
brous but perfectly rigorous "method of exhaustion." In 
determining the ratio of the areas between two curvilinear 
plane figures, say two circles, geometers first inscribed or 
circumscribed similar polygons, and then by increasing in- 
definitely the number of sides, nearly exhausted the spaces 
between the polygons and circumferences. From the theorem 
that similar polygons inscribed in circles are to each other as 
the squares on their diameters, geometers may have divined 
the theorem attributed to Hippocrates of Chios that the 
circles, which differ but little from the last drawn polygons, 
must be to each other as the squares on their diameters. But 
in order to exclude all vagueness and possibility of doubt, 
later Greek geometers applied reasoning like that in Euclid, 
XII. 2, as follows: Let C and c, D and d be respectively the 


circles and diameters in question. Then if the proportion 
D 2 : d 2 = C : c is not true, suppose that D 2 : d 2 = C : d . 
If d < c, then a polygon p can be inscribed in the circle c 
which comes nearer to it in area than does d. If P be the 
corresponding polygon in C, then P : p = D 2 : d 2 = C : d, 
and P : C = p : c' . Since p > c', we have P > C, which is 
absurd. Next they proved by this same method of reductio ad 
absurdum the falsity of the supposition that c' > c. Since d 
can be neither larger nor smaller than c, it must be equal to 
it, Q.E.D. Hankel refers this Method of Exhaustion back to 
Hippocrates of Chios, but the reasons for assigning it to this 
early writer, rather than to Eudoxus, seem insufficient. 

Though progress in geometry at this period is traceable only 
at Athens, yet Ionia, Sicily, Abdera in Thrace, and Cyrene 
produced mathematicians who made creditable contributions 
to the science. We can mention here only Democritus of 
Abdera (about 460-370 B.C.), a pupil of Anaxagoras, a 
friend of Philolaus, and an admirer of the Pythagoreans. He 
visited Egypt and perhaps even Persia. He was a successful 
geometer and wrote on incommensurable lines, on geometry, 
on numbers, and on perspective. None of these works are 
extant. He used to boast that in the construction of plane 
figures with proof no one had yet surpassed him, not even 
the so-called harpedonaptae ("rope-stretchers") of Egypt. By 
this assertion he pays a flattering compliment to the skill and 
ability of the Egyptians. 


The Platonic School. 

During the Peloponnesian War (431-404 B.C.) the progress 
of geometry was checked. After the war, Athens sank into the 
background as a minor political power, but advanced more 
and more to the front as the leader in philosophy, literature, 
and science. Plato was born at Athens in 429 B.C., the year 
of the great plague, and died in 348 B.C. He was a pupil 
and near friend of Socrates, but it was not from him that 
he acquired his taste for mathematics. After the death of 
Socrates, Plato travelled extensively. In Cyrene he studied 
mathematics under Theodorus. He went to Egypt, then to 
Lower Italy and Sicily, where he came in contact with the 
Pythagoreans. Archytas of Tarentum and Timaeus of Locri 
became his intimate friends. On his return to Athens, about 
389 B.C., he founded his school in the groves of the Academia, 
and devoted the remainder of his life to teaching and writing. 

Plato's physical philosophy is partly based on that of 
the Pythagoreans. Like them, he sought in arithmetic and 
geometry the key to the universe. When questioned about the 
occupation of the Deity, Plato answered that "He geometrises 
continually." Accordingly, a knowledge of geometry is a 
necessary preparation for the study of philosophy. To show 
how great a value he put on mathematics and how necessary 
it is for higher speculation, Plato placed the inscription over 
his porch, "Let no one who is unacquainted with geometry 
enter here." Xenocrates, a successor of Plato as teacher in 
the Academy, followed in his master's footsteps, by declining 


to admit a pupil who had no mathematical training, with the 
remark, "Depart, for thou hast not the grip of philosophy." 
Plato observed that geometry trained the mind for correct and 
vigorous thinking. Hence it was that the Eudemian Summary 
says, "He filled his writings with mathematical discoveries, 
and exhibited on every occasion the remarkable connection 
between mathematics and philosophy." 

With Plato as the head-master, we need not wonder 
that the Platonic school produced so large a number of 
mathematicians. Plato did little real original work, but 
he made valuable improvements in the logic and methods 
employed in geometry. It is true that the Sophist geometers 
of the previous century were rigorous in their proofs, but as a 
rule they did not reflect on the inward nature of their methods. 
They used the axioms without giving them explicit expression, 
and the geometrical concepts, such as the point, line, surface, 
etc., without assigning to them formal definitions. The 
Pythagoreans called a point "unity in position," but this is a 
statement of a philosophical theory rather than a definition. 
Plato objected to calling a point a "geometrical fiction." He 
defined a point as the "beginning of a line" or as "an indivisible 
line," and a line as "length without breadth." He called the 
point, line, surface, the 'boundaries' of the line, surface, solid, 
respectively. Many of the definitions in Euclid are to be 
ascribed to the Platonic school. The same is probably true 
of Euclid's axioms. Aristotle refers to Plato the axiom that 
"equals subtracted from equals leave equals." 

One of the greatest achievements of Plato and his school is 


the invention of analysis as a method of proof. To be sure, 
this method had been used unconsciously by Hippocrates 
and others; but Plato, like a true philosopher, turned the 
instinctive logic into a conscious, legitimate method. 

The terms synthesis and analysis are used in mathematics in 
a more special sense than in logic. In ancient mathematics they 
had a different meaning from what they now have. The oldest 
definition of mathematical analysis as opposed to synthesis 
is that given in Euclid, XIII. 5, which in all probability was 
framed by Eudoxus: "Analysis is the obtaining of the thing 
sought by assuming it and so reasoning up to an admitted 
truth; synthesis is the obtaining of the thing sought by 
reasoning up to the inference and proof of it." The analytic 
method is not conclusive, unless all operations involved in 
it are known to be reversible. To remove all doubt, the 
Greeks, as a rule, added to the analytic process a synthetic 
one, consisting of a reversion of all operations occurring in the 
analysis. Thus the aim of analysis was to aid in the discovery 
of synthetic proofs or solutions. 

Plato is said to have solved the problem of the duplication of 
the cube. But the solution is open to the very same objection 
which he made to the solutions by Archytas, Eudoxus, and 
Menaechmus. He called their solutions not geometrical, but 
mechanical, for they required the use of other instruments 
than the ruler and compasses. He said that thereby "the good 
of geometry is set aside and destroyed, for we again reduce 
it to the world of sense, instead of elevating and imbuing it 
with the eternal and incorporeal images of thought, even as 


it is employed by God, for which reason He always is God." 
These objections indicate either that the solution is wrongly 
attributed to Plato or that he wished to show how easily 
non-geometric solutions of that character can be found. It is 
now generally admitted that the duplication problem, as well 
as the trisection and quadrature problems, cannot be solved 
by means of the ruler and compass only. 

Plato gave a healthful stimulus to the study of stereometry, 
which until his time had been entirely neglected. The sphere 
and the regular solids had been studied to some extent, but 
the prism, pyramid, cylinder, and cone were hardly known to 
exist. All these solids became the subjects of investigation 
by the Platonic school. One result of these inquiries was 
epoch-making. Menaechmus, an associate of Plato and 
pupil of Eudoxus, invented the conic sections, which, in course 
of only a century, raised geometry to the loftiest height which 
it was destined to reach during antiquity. Menaechmus cut 
three kinds of cones, the 'right-angled,' 'acute-angled,' and 
'obtuse-angled,' by planes at right angles to a side of the 
cones, and thus obtained the three sections which we now 
call the parabola, ellipse, and hyperbola. Judging from 
the two very elegant solutions of the "Delian Problem" by 
means of intersections of these curves, Menaechmus must have 
succeeded well in investigating their properties. 

Another great geometer was Dinostratus, the brother of 
Menaechmus and pupil of Plato. Celebrated is his mechanical 
solution of the quadrature of the circle, by means of the 
quadratrix of Hippias. 


Perhaps the most brilliant mathematician of this period was 
Eudoxus. He was born at Cnidus about 408 B.C., studied 
under Archytas, and later, for two months, under Plato. He 
was imbued with a true spirit of scientific inquiry, and has 
been called the father of scientific astronomical observation. 
From the fragmentary notices of his astronomical researches, 
found in later writers, Ideler and Schiaparelli succeeded in 
reconstructing the system of Eudoxus with its celebrated 
representation of planetary motions by "concentric spheres." 
Eudoxus had a school at Cyzicus, went with his pupils to 
Athens, visiting Plato, and then returned to Cyzicus, where 
he died 355 B.C. The fame of the academy of Plato is to a 
large extent due to Eudoxus's pupils of the school at Cyzicus, 
among whom are Menaechmus, Dinostratus, Athenaeus, and 
Helicon. Diogenes Laertius describes Eudoxus as astronomer, 
physician, legislator, as well as geometer. The Eudemian 
Summary says that Eudoxus "first increased the number of 
general theorems, added to the three proportions three more, 
and raised to a considerable quantity the learning, begun by 
Plato, on the subject of the section, to which he applied the 
analytical method." By this 'section' is meant, no doubt, the 
"golden section" (sectio aurea), which cuts a line in extreme 
and mean ratio. The first five propositions in Euclid XIII. 
relate to lines cut by this section, and are generally attributed 
to Eudoxus. Eudoxus added much to the knowledge of solid 
geometry. He proved, says Archimedes, that a pyramid is 
exactly one-third of a prism, and a cone one-third of a cylinder, 
having equal base and altitude. The proof that spheres are to 


each other as the cubes of their radii is probably due to him. 
He made frequent and skilful use of the method of exhaustion, 
of which he was in all probability the inventor. A scholiast 
on Euclid, thought to be Proclus, says further that Eudoxus 
practically invented the whole of Euclid's fifth book. Eudoxus 
also found two mean proportionals between two given lines, 
but the method of solution is not known. 

Plato has been called a maker of mathematicians. Besides 
the pupils already named, the Eudemian Summary mentions 
the following: Theaetetus of Athens, a man of great natural 
gifts, to whom, no doubt, Euclid was greatly indebted in 
the composition of the 10th book, [8] treating of incommen- 
surables; Leodamas of Thasos; Neocleides and his pupil 
Leon, who added much to the work of their predecessors, 
for Leon wrote an Elements carefully designed, both in num- 
ber and utility of its proofs; Theudius of Magnesia, who 
composed a very good book of Elements and generalised 
propositions, which had been confined to particular cases; 
Hermotimus of Colophon, who discovered many proposi- 
tions of the Elements and composed some on loci] and, finally, 
the names of Amyclas of Heraclea, Cyzicenus of Athens, 
and Philippus of Mende. 

A skilful mathematician of whose life and works we have no 
details is Aristaeus, the elder, probably a senior contemporary 
of Euclid. The fact that he wrote a work on conic sections 
tends to show that much progress had been made in their 
study during the time of Menaechmus. Aristaeus wrote also 
on regular solids and cultivated the analytic method. His 


works contained probably a summary of the researches of the 
Platonic school. [8] 

Aristotle (384-322 B.C.), the systematiser of deductive 
logic, though not a professed mathematician, promoted the 
science of geometry by improving some of the most difficult 
definitions. His Physics contains passages with suggestive 
hints of the principle of virtual velocities. About his time there 
appeared a work called Mechanica, of which he is regarded by 
some as the author. Mechanics was totally neglected by the 
Platonic school. 

The First Alexandrian School. 

In the previous pages we have seen the birth of geometry in 
Egypt, its transference to the Ionian Islands, thence to Lower 
Italy and to Athens. We have witnessed its growth in Greece 
from feeble childhood to vigorous manhood, and now we shall 
see it return to the land of its birth and there derive new 

During her declining years, immediately following the 
Peloponnesian War, Athens produced the greatest scientists 
and philosophers of antiquity. It was the time of Plato and 
Aristotle. In 338 B.C., at the battle of Chaeronea, Athens 
was beaten by Philip of Macedon, and her power was broken 
forever. Soon after, Alexander the Great, the son of Philip, 
started out to conquer the world. In eleven years he built 
up a great empire which broke to pieces in a day. Egypt 
fell to the lot of Ptolemy Soter. Alexander had founded the 


seaport of Alexandria, which soon became "the noblest of all 
cities." Ptolemy made Alexandria the capital. The history of 
Egypt during the next three centuries is mainly the history 
of Alexandria. Literature, philosophy, and art were diligently 
cultivated. Ptolemy created the university of Alexandria. He 
founded the great Library and built laboratories, museums, a 
zoological garden, and promenades. Alexandria soon became 
the great centre of learning. 

Demetrius Phalereus was invited from Athens to take charge 
of the Library, and it is probable, says Gow, that Euclid was 
invited with him to open the mathematical school. Euclid's 
greatest activity was during the time of the first Ptolemy, 
who reigned from 306 to 283 B.C. Of the life of Euclid, 
little is known, except what is added by Proclus to the 
Eudemian Summary. Euclid, says Proclus, was younger 
than Plato and older than Eratosthenes and Archimedes, the 
latter of whom mentions him. He was of the Platonic sect, 
and well read in its doctrines. He collected the Elements, 
put in order much that Eudoxus had prepared, completed 
many things of Theaetetus, and was the first who reduced 
to unobjectionable demonstration the imperfect attempts of 
his predecessors. When Ptolemy once asked him if geometry 
could not be mastered by an easier process than by studying 
the Elements, Euclid returned the answer, "There is no 
royal road to geometry." Pappus states that Euclid was 
distinguished by the fairness and kindness of his disposition, 
particularly toward those who could do anything to advance 
the mathematical sciences. Pappus is evidently making a 


contrast to Apollonius, of whom he more than insinuates 
the opposite character. [9] A pretty little story is related by 
Stobaeus: [6] "A youth who had begun to read geometry with 
Euclid, when he had learnt the first proposition, inquired, 
'What do I get by learning these things?' So Euclid called 
his slave and said, 'Give him threepence, since he must make 
gain out of what he learns. ' " These are about all the personal 
details preserved by Greek writers. Syrian and Arabian 
writers claim to know much more, but they are unreliable. 
At one time Euclid of Alexandria was universally confounded 
with Euclid of Megara, who lived a century earlier. 

The fame of Euclid has at all times rested mainly upon his 
book on geometry, called the Elements. This book was so far 
superior to the Elements written by Hippocrates, Leon, and 
Theudius, that the latter works soon perished in the struggle 
for existence. The Greeks gave Euclid the special title of 
"the author of the Elements." It is a remarkable fact in the 
history of geometry, that the Elements of Euclid, written two 
thousand years ago, are still regarded by many as the best 
introduction to the mathematical sciences. In England they 
are used at the present time extensively as a text-book in 
schools. Some editors of Euclid have, however, been inclined 
to credit him with more than is his due. They would have us 
believe that a finished and unassailable system of geometry 
sprang at once from the brain of Euclid, "an armed Minerva 
from the head of Jupiter." They fail to mention the earlier 
eminent mathematicians from whom Euclid got his material. 
Comparatively few of the propositions and proofs in the 


Elements are his own discoveries. In fact, the proof of the 
"Theorem of Pythagoras" is the only one directly ascribed 
to him. Allman conjectures that the substance of Books I., 
II., IV. comes from the Pythagoreans, that the substance 
of Book VI. is due to the Pythagoreans and Eudoxus, the 
latter contributing the doctrine of proportion as applicable 
to incommensurables and also the Method of Exhaustions 
(Book XII.), that Theaetetus contributed much toward Books 
X. and XIII., that the principal part of the original work 
of Euclid himself is to be found in Book X. [8] Euclid was 
the greatest systematiser of his time. By careful selection 
from the material before him, and by logical arrangement of 
the propositions selected, he built up, from a few definitions 
and axioms, a proud and lofty structure. It would be 
erroneous to believe that he incorporated into his Elements 
all the elementary theorems known at his time. Archimedes, 
Apollonius, and even he himself refer to theorems not included 
in his Elements, as being well-known truths. 

The text of the Elements now commonly used is Theon's 
edition. Theon of Alexandria, the father of Hypatia, brought 
out an edition, about 700 years after Euclid, with some 
alterations in the text. As a consequence, later commentators, 
especially Robert Simson, who laboured under the idea that 
Euclid must be absolutely perfect, made Theon the scapegoat 
for all the defects which they thought they could discover in 
the text as they knew it. But among the manuscripts sent by 
Napoleon I. from the Vatican to Paris was found a copy of the 
Elements believed to be anterior to Theon's recension. Many 


variations from Theon's version were noticed therein, but they 
were not at all important, and showed that Theon generally 
made only verbal changes. The defects in the Elements for 
which Theon was blamed must, therefore, be due to Euclid 
himself. The Elements has been considered as offering models 
of scrupulously rigorous demonstrations. It is certainly true 
that in point of rigour it compares favourably with its modern 
rivals; but when examined in the light of strict mathematical 
logic, it has been pronounced by C. S. Peirce to be "riddled 
with fallacies." The results are correct only because the 
writer's experience keeps him on his guard. 

At the beginning of our editions of the Elements, under the 
head of definitions, are given the assumptions of such notions 
as the point, line, etc., and some verbal explanations. Then 
follow three postulates or demands, and twelve axioms. The 
term 'axiom' was used by Proclus, but not by Euclid. He 
speaks, instead, of 'common notions' — common either to all 
men or to all sciences. There has been much controversy among 
ancient and modern critics on the postulates and axioms. An 
immense preponderance of manuscripts and the testimony of 
Proclus place the 'axioms' about right angles and parallels 
(Axioms 11 and 12) among the postulates. [9, 10] This is 
indeed their proper place, for they are really assumptions, and 
not common notions or axioms. The postulate about parallels 
plays an important role in the history of non-Euclidean 
geometry. The only postulate which Euclid missed was the 
one of superposition, according to which figures can be moved 
about in space without any alteration in form or magnitude. 


The Elements contains thirteen books by Euclid, and two, 
of which it is supposed that Hypsicles and Damascius are the 
authors. The first four books are on plane geometry. The 
fifth book treats of the theory of proportion as applied to 
magnitudes in general. The sixth book develops the geometry 
of similar figures. The seventh, eighth, ninth books are on the 
theory of numbers, or on arithmetic. In the ninth book is found 
the proof to the theorem that the number of primes is infinite. 
The tenth book treats of the theory of incommensurables. The 
next three books are on stereometry. The eleventh contains its 
more elementary theorems; the twelfth, the metrical relations 
of the pyramid, prism, cone, cylinder, and sphere. The 
thirteenth treats of the regular polygons, especially of the 
triangle and pentagon, and then uses them as faces of the five 
regular solids; namely, the tetraedron, octaedron, icosaedron, 
cube, and dodecaedron. The regular solids were studied so 
extensively by the Platonists that they received the name of 
"Platonic figures." The statement of Proclus that the whole 
aim of Euclid in writing the Elements was to arrive at the 
construction of the regular solids, is obviously wrong. The 
fourteenth and fifteenth books, treating of solid geometry, are 

A remarkable feature of Euclid's, and of all Greek geometry 
before Archimedes is that it eschews mensuration. Thus the 
theorem that the area of a triangle equals half the product of 
its base and its altitude is foreign to Euclid. 

Another extant book of Euclid is the Data. It seems to have 
been written for those who, having completed the Elements, 


wish to acquire the power of solving new problems proposed 
to them. The Data is a course of practice in analysis. It 
contains little or nothing that an intelligent student could not 
pick up from the Elements itself. Hence it contributes little to 
the stock of scientific knowledge. The following are the other 
extant works generally attributed to Euclid: Phenomena, a 
work on spherical geometry and astronomy; Optics, which 
develops the hypothesis that light proceeds from the eye, and 
not from the object seen; Catoptrica, containing propositions 
on reflections from mirrors; De Divisionibus, a treatise on 
the division of plane figures into parts having to one another 
a given ratio; Sectio Canonis, a work on musical intervals. 
His treatise on Porisms is lost; but much learning has been 
expended by Robert Simson and M. Chasles in restoring it 
from numerous notes found in the writings of Pappus. The 
term 'porism' is vague in meaning. The aim of a porism is not 
to state some property or truth, like a theorem, nor to effect a 
construction, like a problem, but to find and bring to view a 
thing which necessarily exists with given numbers or a given 
construction, as, to find the centre of a given circle, or to find 
the G.C.D. of two given numbers. [6] His other lost works are 
Fallacies, containing exercises in detection of fallacies; Conic 
Sections, in four books, which are the foundation of a work on 
the same subject by Apollonius; and Loci on a Surface, the 
meaning of which title is not understood. Heiberg believes it 
to mean "loci which are surfaces." 

The immediate successors of Euclid in the mathematical 
school at Alexandria were probably Conon, Dositheus, and 


Zeuxippus, but little is known of them. 

Archimedes (2877-212 B.C.), the greatest mathematician 
of antiquity, was born in Syracuse. Plutarch calls him a 
relation of King Hieron; but more reliable is the statement 
of Cicero, who tells us he was of low birth. Diodorus says 
he visited Egypt, and, since he was a great friend of Conon 
and Eratosthenes, it is highly probable that he studied in 
Alexandria. This belief is strengthened by the fact that he had 
the most thorough acquaintance with all the work previously 
done in mathematics. He returned, however, to Syracuse, 
where he made himself useful to his admiring friend and 
patron, King Hieron, by applying his extraordinary inventive 
genius to the construction of various war-engines, by which 
he inflicted much loss on the Romans during the siege of 
Marcellus. The story that, by the use of mirrors reflecting the 
sun's rays, he set on fire the Roman ships, when they came 
within bow-shot of the walls, is probably a fiction. The city 
was taken at length by the Romans, and Archimedes perished 
in the indiscriminate slaughter which followed. According to 
tradition, he was, at the time, studying the diagram to some 
problem drawn in the sand. As a Roman soldier approached 
him, he called out, "Don't spoil my circles." The soldier, 
feeling insulted, rushed upon him and killed him. No blame 
attaches to the Roman general Marcellus, who admired his 
genius, and raised in his honour a tomb bearing the figure of a 
sphere inscribed in a cylinder. When Cicero was in Syracuse, 
he found the tomb buried under rubbish. 

Archimedes was admired by his fellow-citizens chiefly for 


his mechanical inventions; he himself prized far more highly 
his discoveries in pure science. He declared that "every kind 
of art which was connected with daily needs was ignoble and 
vulgar." Some of his works have been lost. The following are 
the extant books, arranged approximately in chronological 
order: 1. Two books on Equiponderance of Planes or Centres 
of Plane Gravities, between which is inserted his treatise 
on the Quadrature of the Parabola; 2. Two books on the 
Sphere and Cylinder; 3. The Measurement of the Circle; 4. On 
Spirals; 5. Conoids and Spheroids; 6. The Sand- Counter; 
7. Two books on Floating Bodies; 8. Fifteen Lemmas. 

In the book on the Measurement of the Circle, Archimedes 
proves first that the area of a circle is equal to that of a right 
triangle having the length of the circumference for its base, 
and the radius for its altitude. In this he assumes that there 
exists a straight line equal in length to the circumference — an 
assumption objected to by some ancient critics, on the ground 
that it is not evident that a straight line can equal a curved 
one. The finding of such a line was the next problem. He 
first finds an upper limit to the ratio of the circumference to 
the diameter, or ir. To do this, he starts with an equilateral 
triangle of which the base is a tangent and the vertex is the 
centre of the circle. By successively bisecting the angle at 
the centre, by comparing ratios, and by taking the irrational 
square roots always a little too small, he finally arrived at 
the conclusion that n < 3y. Next he finds a lower limit 
by inscribing in the circle regular polygons of 6, 12, 24, 48, 
96 sides, finding for each successive polygon its perimeter, 


which is, of course, always less than the circumference. Thus 
he finally concludes that "the circumference of a circle exceeds 
three times its diameter by a part which is less than 7 but 
more than ij of the diameter." This approximation is exact 
enough for most purposes. 

The Quadrature of the Parabola contains two solutions to 
the problem — one mechanical, the other geometrical. The 
method of exhaustion is used in both. 

Archimedes studied also the ellipse and accomplished its 
quadrature, but to the hyperbola he seems to have paid less 
attention. It is believed that he wrote a book on conic sections. 

Of all his discoveries Archimedes prized most highly those 
in his Sphere and Cylinder. In it are proved the new theorems, 
that the surface of a sphere is equal to four times a great 
circle; that the surface of a segment of a sphere is equal to a 
circle whose radius is the straight line drawn from the vertex 
of the segment to the circumference of its basal circle; that 
the volume and the surface of a sphere are | of the volume 
and surface, respectively, of the cylinder circumscribed about 
the sphere. Archimedes desired that the figure to the last 
proposition be inscribed on his tomb. This was ordered done 
by Marcellus. 

The spiral now called the "spiral of Archimedes," and de- 
scribed in the book On Spirals, was discovered by Archimedes, 
and not, as some believe, by his friend Conon. [3] His treatise 
thereon is, perhaps, the most wonderful of all his works. 
Nowadays, subjects of this kind are made easy by the use 
of the infinitesimal calculus. In its stead the ancients used 


the method of exhaustion. Nowhere is the fertility of his 
genius more grandly displayed than in his masterly use of this 
method. With Euclid and his predecessors the method of 
exhaustion was only the means of proving propositions which 
must have been seen and believed before they were proved. 
But in the hands of Archimedes it became an instrument of 
discovery. [9] 

By the word 'conoid,' in his book on Conoids and Spheroids, 
is meant the solid produced by the revolution of a parabola 
or a hyperbola about its axis. Spheroids are produced by the 
revolution of an ellipse, and are long or flat, according as the 
ellipse revolves around the major or minor axis. The book 
leads up to the cubature of these solids. 

We have now reviewed briefly all his extant works on 
geometry. His arithmetical treatise and problems will be 
considered later. We shall now notice his works on mechanics. 
Archimedes is the author of the first sound knowledge on 
this subject. Archytas, Aristotle, and others attempted 
to form the known mechanical truths into a science, but 
failed. Aristotle knew the property of the lever, but could 
not establish its true mathematical theory. The radical and 
fatal defect in the speculations of the Greeks, says Whewell, 
was "that though they had in their possession facts and ideas, 
the ideas were not distinct and appropriate to the facts." For 
instance, Aristotle asserted that when a body at the end of a 
lever is moving, it may be considered as having two motions; 
one in the direction of the tangent and one in the direction 
of the radius; the former motion is, he says, according to 


nature, the latter contrary to nature. These inappropriate 
notions of 'natural' and 'unnatural' motions, together with 
the habits of thought which dictated these speculations, made 
the perception of the true grounds of mechanical properties 
impossible. [11] It seems strange that even after Archimedes 
had entered upon the right path, this science should have 
remained absolutely stationary till the time of Galileo — a 
period of nearly two thousand years. 

The proof of the property of the lever, given in his 
Equiponderance of Planes, holds its place in text-books to this 
day. His estimate of the efficiency of the lever is expressed in 
the saying attributed to him, "Give me a fulcrum on which to 
rest, and I will move the earth." 

While the Equiponderance treats of solids, or the equilibrium 
of solids, the book on Floating Bodies treats of hydrostatics. 
His attention was first drawn to the subject of specific gravity 
when King Hieron asked him to test whether a crown, professed 
by the maker to be pure gold, was not alloyed with silver. 
The story goes that our philosopher was in a bath when the 
true method of solution flashed on his mind. He immediately 
ran home, naked, shouting, "I have found it!" To solve the 
problem, he took a piece of gold and a piece of silver, each 
weighing the same as the crown. According to one author, 
he determined the volume of water displaced by the gold, 
silver, and crown respectively, and calculated from that the 
amount of gold and silver in the crown. According to another 
writer, he weighed separately the gold, silver, and crown, 
while immersed in water, thereby determining their loss of 


weight in water. From these data he easily found the solution. 
It is possible that Archimedes solved the problem by both 

After examining the writings of Archimedes, one can well 
understand how, in ancient times, an 'Archimedean problem' 
came to mean a problem too deep for ordinary minds to solve, 
and how an 'Archimedean proof came to be the synonym for 
unquestionable certainty. Archimedes wrote on a very wide 
range of subjects, and displayed great profundity in each. He 
is the Newton of antiquity. 

Eratosthenes, eleven years younger than Archimedes, was 
a native of Cyrene. He was educated in Alexandria under 
Callimachus the poet, whom he succeeded as custodian of the 
Alexandrian Library. His many-sided activity may be inferred 
from his works. He wrote on Good and Evil, Measurement of 
the Earth, Comedy, Geography, Chronology, Constellations, 
and the Duplication of the Cube. He was also a philologian and 
a poet. He measured the obliquity of the ecliptic and invented 
a device for finding prime numbers. Of his geometrical 
writings we possess only a letter to Ptolemy Euergetes, giving 
a history of the duplication problem and also the description 
of a very ingenious mechanical contrivance of his own to solve 
it. In his old age he lost his eyesight, and on that account is 
said to have committed suicide by voluntary starvation. 

About forty years after Archimedes flourished Apollonius 
of Perga, whose genius nearly equalled that of his great 
predecessor. He incontestably occupies the second place 
in distinction among ancient mathematicians. Apollonius 


was born in the reign of Ptolemy Euergetes and died under 
Ptolemy Philopator, who reigned 222-205 B.C. He studied at 
Alexandria under the successors of Euclid, and for some time, 
also, at Pergamum, where he made the acquaintance of that 
Eudemus to whom he dedicated the first three books of his 
Conic Sections. The brilliancy of his great work brought him 
the title of the "Great Geometer." This is all that is known of 
his life. 

His Conic Sections were in eight books, of which the first 
four only have come down to us in the original Greek. The 
next three books were unknown in Europe till the middle of 
the seventeenth century, when an Arabic translation, made 
about 1250, was discovered. The eighth book has never 
been found. In 1710 Halley of Oxford published the Greek 
text of the first four books and a Latin translation of the 
remaining three, together with his conjectural restoration 
of the eighth book, founded on the introductory lemmas of 
Pappus. The first four books contain little more than the 
substance of what earlier geometers had done. Eutocius 
tells us that Heraclides, in his life of Archimedes, accused 
Apollonius of having appropriated, in his Conic Sections, 
the unpublished discoveries of that great mathematician. 
It is difficult to believe that this charge rests upon good 
foundation. Eutocius quotes Geminus as replying that 
neither Archimedes nor Apollonius claimed to have invented 
the conic sections, but that Apollonius had introduced a 
real improvement. While the first three or four books were 
founded on the works of Menaechmus, Aristaeus, Euclid, and 


Archimedes, the remaining ones consisted almost entirely of 
new matter. The first three books were sent to Eudemus 
at intervals, the other books (after Eudemus 's death) to one 
Attalus. The preface of the second book is interesting as 
showing the mode in which Greek books were 'published' at 
this time. It reads thus: "I have sent my son Apollonius to 
bring you (Eudemus) the second book of my Conies. Read it 
carefully and communicate it to such others as are worthy of 
it. If Philonides, the geometer, whom I introduced to you at 
Ephesus, comes into the neighbourhood of Pergamum, give it 
to him also." [12] 

The first book, says Apollonius in his preface to it, "contains 
the mode of producing the three sections and the conjugate 
hyperbolas and their principal characteristics, more fully and 
generally worked out than in the writings of other authors." 
We remember that Menaechmus, and all his successors down to 
Apollonius, considered only sections of right cones by a plane 
perpendicular to their sides, and that the three sections were 
obtained each from a different cone. Apollonius introduced 
an important generalisation. He produced all the sections 
from one and the same cone, whether right or scalene, and 
by sections which may or may not be perpendicular to its 
sides. The old names for the three curves were now no longer 
applicable. Instead of calling the three curves, sections of the 
'acute- angled,' 'right-angled,' and 'obtuse-angled' cone, he 
called them ellipse, parabola, and hyperbola, respectively. To 
be sure, we find the words 'parabola' and 'ellipse' in the works 
of Archimedes, but they are probably only interpolations. 


The word 'ellipse' was applied because y 2 < px, p being 
the parameter; the word 'parabola' was introduced because 
y 2 = px, and the term 'hyperbola' because y 2 > px. 

The treatise of Apollonius rests on a unique property of 
conic sections, which is derived directly from the nature of 
the cone in which these sections are found. How this property 
forms the key to the system of the ancients is told in a masterly 
way by M. Chasles. [13] "Conceive," says he, "an oblique cone 
on a circular base; the straight line drawn from its summit to 
the centre of the circle forming its base is called the axis of 
the cone. The plane passing through the axis, perpendicular 
to its base, cuts the cone along two lines and determines in 
the circle a diameter; the triangle having this diameter for 
its base and the two lines for its sides, is called the triangle 
through the axis. In the formation of his conic sections, 
Apollonius supposed the cutting plane to be perpendicular 
to the plane of the triangle through the axis. The points 
in which this plane meets the two sides of this triangle are 
the vertices of the curve; and the straight line which joins 
these two points is a diameter of it. Apollonius called this 
diameter latus transversum. At one of the two vertices of 
the curve erect a perpendicular (latus rectum) to the plane 
of the triangle through the axis, of a certain length, to be 
determined as we shall specify later, and from the extremity of 
this perpendicular draw a straight line to the other vertex of 
the curve; now, through any point whatever of the diameter of 
the curve, draw at right angles an ordinate: the square of this 
ordinate, comprehended between the diameter and the curve, 


will be equal to the rectangle constructed on the portion of 
the ordinate comprised between the diameter and the straight 
line, and the part of the diameter comprised between the first 
vertex and the foot of the ordinate. Such is the characteristic 
property which Apollonius recognises in his conic sections and 
which he uses for the purpose of inferring from it, by adroit 
transformations and deductions, nearly all the rest. It plays, 
as we shall see, in his hands, almost the same role as the 
equation of the second degree with two variables (abscissa and 
ordinate) in the system of analytic geometry of Descartes. 

"It will be observed from this that the diameter of the 
curve and the perpendicular erected at one of its extremities 
suffice to construct the curve. These are the two elements 
which the ancients used, with which to establish their theory 
of conies. The perpendicular in question was called by them 
latus erectum; the moderns changed this name first to that of 
latus rectum, and afterwards to that oi parameter." 

The first book of the Conic Sections of Apollonius is almost 
wholly devoted to the generation of the three principal conic 

The second book treats mainly of asymptotes, axes, and 

The third book treats of the equality or proportionality 
of triangles, rectangles, or squares, of which the component 
parts are determined by portions of transversals, chords, 
asymptotes, or tangents, which are frequently subject to a 
great number of conditions. It also touches the subject of foci 
of the ellipse and hyperbola. 


In the fourth book, Apollonius discusses the harmonic 
division of straight lines. He also examines a system of two 
conies, and shows that they cannot cut each other in more 
than four points. He investigates the various possible relative 
positions of two conies, as, for instance, when they have one 
or two points of contact with each other. 

The fifth book reveals better than any other the giant 
intellect of its author. Difficult questions of maxima and 
minima, of which few examples are found in earlier works, are 
here treated most exhaustively. The subject investigated is, 
to find the longest and shortest lines that can be drawn from 
a given point to a conic. Here are also found the germs of the 
subject of evolutes and centres of osculation. 

The sixth book is on the similarity of conies. 

The seventh book is on conjugate diameters. 

The eighth book, as restored by H alley, continues the 
subject of conjugate diameters. 

It is worthy of notice that Apollonius nowhere introduces 
the notion of directrix for a conic, and that, though he 
incidentally discovered the focus of an ellipse and hyperbola, 
he did not discover the focus of a parabola. [6] Conspicuous 
in his geometry is also the absence of technical terms and 
symbols, which renders the proofs long and cumbrous. 

The discoveries of Archimedes and Apollonius, says M. 
Chasles, [13] marked the most brilliant epoch of ancient 
geometry. Two questions which have occupied geometers of 
all periods may be regarded as having originated with them. 


The first of these is the quadrature of curvilinear figures, 
which gave birth to the infinitesimal calculus. The second 
is the theory of conic sections, which was the prelude to the 
theory of geometrical curves of all degrees, and to that portion 
of geometry which considers only the forms and situations of 
figures, and uses only the intersection of lines and surfaces and 
the ratios of rectilineal distances. These two great divisions 
of geometry may be designated by the names of Geometry 
of Measurements and Geometry of Forms and Situations, or, 
Geometry of Archimedes and of Apollonius. 

Besides the Conic Sections, Pappus ascribes to Apollonius 
the following works: On Contacts, Plane Loci, Inclinations, 
Section of an Area, Determinate Section, and gives lemmas 
from which attempts have been made to restore the lost 
originals. Two books on De Sectione Rationis have been 
found in the Arabic. The book on Contacts, as restored by 
Vieta, contains the so-called "Apollonian Problem": Given 
three circles, to find a fourth which shall touch the three. 

Euclid, Archimedes, and Apollonius brought geometry 
to as high a state of perfection as it perhaps could be 
brought without first introducing some more general and 
more powerful method than the old method of exhaustion. 
A briefer symbolism, a Cartesian geometry, an infinitesimal 
calculus, were needed. The Greek mind was not adapted 
to the invention of general methods. Instead of a climb to 
still loftier heights we observe, therefore, on the part of later 
Greek geometers, a descent, during which they paused here 
and there to look around for details which had been passed by 


in the hasty ascent. [3] 

Among the earliest successors of Apollonius was Nico- 
medes. Nothing definite is known of him, except that he 
invented the conchoid ("mussel-like"). He devised a little 
machine by which the curve could be easily described. With 
aid of the conchoid he duplicated the cube. The curve can 
also be used for trisecting angles in a way much resembling 
that in the eighth lemma of Archimedes. Proclus ascribes this 
mode of trisection to Nicomedes, but Pappus, on the other 
hand, claims it as his own. The conchoid was used by Newton 
in constructing curves of the third degree. 

About the time of Nicomedes, flourished also Diocles, the 
inventor of the cissoid ("ivy-like"). This curve he used for 
finding two mean proportionals between two given straight 

About the life of Perseus we know as little as about that of 
Nicomedes and Diocles. He lived some time between 200 and 
100 B.C. From Heron and Geminus we learn that he wrote a 
work on the spire, a sort of anchor-ring surface described by 
Heron as being produced by the revolution of a circle around 
one of its chords as an axis. The sections of this surface yield 
peculiar curves called spiral sections, which, according to 
Geminus, were thought out by Perseus. These curves appear 
to be the same as the Hippopede of Eudoxus. 

Probably somewhat later than Perseus lived Zenodorus. 
He wrote an interesting treatise on a new subject; namely, 
is operimetrical figures. Fourteen propositions are preserved by 
Pappus and Theon. Here are a few of them: Ofisoperimetrical, 


regular polygons, the one having the largest number of angles 
has the greatest area; the circle has a greater area than 
any regular polygon of equal periphery; of all isoperimetrical 
polygons of n sides, the regular is the greatest; of all solids 
having surfaces equal in area, the sphere has the greatest 

Hypsicles (between 200 and 100 B.C.) was supposed to 
be the author of both the fourteenth and fifteenth books of 
Euclid, but recent critics are of opinion that the fifteenth book 
was written by an author who lived several centuries after 
Christ. The fourteenth book contains seven elegant theorems 
on regular solids. A treatise of Hypsicles on Risings is of 
interest because it is the first Greek work giving the division 
of the circumference into 360 degrees after the fashion of the 

Hipparchus of Nicaaa in Bithynia was the greatest as- 
tronomer of antiquity. He established inductively the famous 
theory of epicycles and eccentrics. As might be expected, he 
was interested in mathematics, not per se, but only as an aid 
to astronomical inquiry. No mathematical writings of his are 
extant, but Theon of Alexandria informs us that Hipparchus 
originated the science of trigonometry, and that he calcu- 
lated a "table of chords" in twelve books. Such calculations 
must have required a ready knowledge of arithmetical and 
algebraical operations. 

About 100 B.C. flourished Heron the Elder of Alexandria. 
He was the pupil of Ctesibius, who was celebrated for his 
ingenious mechanical inventions, such as the hydraulic organ, 


the water-clock, and catapult. It is believed by some that 
Heron was a son of Ctesibius. He exhibited talent of the same 
order as did his master by the invention of the eolipile and 
a curious mechanism known as "Heron's fountain." Great 
uncertainty exists concerning his writings. Most authorities 
believe him to be the author of an important Treatise on 
the Dioptra, of which there exist three manuscript copies, 
quite dissimilar. But M. Marie [14] thinks that the Dioptra is 
the work of Heron the Younger, who lived in the seventh or 
eighth century after Christ, and that Geodesy, another book 
supposed to be by Heron, is only a corrupt and defective copy 
of the former work. Dioptra contains the important formula 
for finding the area of a triangle expressed in terms of its sides; 
its derivation is quite laborious and yet exceedingly ingenious. 
"It seems to me difficult to believe," says Chasles, "that so 
beautiful a theorem should be found in a work so ancient as 
that of Heron the Elder, without that some Greek geometer 
should have thought to cite it." Marie lays great stress on 
this silence of the ancient writers, and argues from it that 
the true author must be Heron the Younger or some writer 
much more recent than Heron the Elder. But no reliable 
evidence has been found that there actually existed a second 
mathematician by the name of Heron. 

"Dioptra," says Venturi, were instruments which had great 
resemblance to our modern theodolites. The book Dioptra is 
a treatise on geodesy containing solutions, with aid of these 
instruments, of a large number of questions in geometry, such 
as to find the distance between two points, of which one only 


is accessible, or between two points which are visible but both 
inaccessible; from a given point to draw a perpendicular to a 
line which cannot be approached; to find the difference of level 
between two points; to measure the area of a field without 
entering it. 

Heron was a practical surveyor. This may account for the 
fact that his writings bear so little resemblance to those of 
the Greek authors, who considered it degrading the science to 
apply geometry to surveying. The character of his geometry 
is not Grecian, but decidedly Egyptian. This fact is the 
more surprising when we consider that Heron demonstrated 
his familiarity with Euclid by writing a commentary on the 
Elements. [21] Some of Heron's formulas point to an old 
Egyptian origin. Thus, besides the above exact formula for 
the area of a triangle in terms of its sides, Heron gives the 

formula x -, which bears a striking likeness to the 

2 2' 6 

formula x for finding the area of a quadrangle, 

found in the Edfu inscriptions. There are, moreover, points of 
resemblance between Heron's writings and the ancient Ahmes 
papyrus. Thus Ahmes used unit-fractions exclusively; Heron 
uses them oftener than other fractions. Like Ahmes and 
the priests at Edfu, Heron divides complicated figures into 
simpler ones by drawing auxiliary lines; like them, he shows, 
throughout, a special fondness for the isosceles trapezoid. 

The writings of Heron satisfied a practical want, and for 
that reason were borrowed extensively by other peoples. We 
find traces of them in Rome, in the Occident during the Middle 


Ages, and even in India. 

Geminus of Rhodes (about 70 B.C.) published an astro- 
nomical work still extant. He wrote also a book, now lost, 
on the Arrangement of Mathematics, which contained many 
valuable notices of the early history of Greek mathematics. 
Proclus and Eutocius quote it frequently. Theodosius of 
Tripolis is the author of a book of little merit on the geometry 
of the sphere. Dionysodorus of Amisus in Pontus applied 
the intersection of a parabola and hyperbola to the solution 
of a problem which Archimedes, in his Sphere and Cylinder, 
had left incomplete. The problem is "to cut a sphere so that 
its segments shall be in a given ratio." 

We have now sketched the progress of geometry down to 
the time of Christ. Unfortunately, very little is known of the 
history of geometry between the time of Apollonius and the 
beginning of the Christian era. The names of quite a number 
of geometers have been mentioned, but very few of their works 
are now extant. It is certain, however, that there were no 
mathematicians of real genius from Apollonius to Ptolemy, 
excepting Hipparchus and perhaps Heron. 

The Second Alexandrian School. 

The close of the dynasty of the Lagides which ruled Egypt 
from the time of Ptolemy Soter, the builder of Alexandria, for 
300 years; the absorption of Egypt into the Roman Empire; 
the closer commercial relations between peoples of the East 
and of the West; the gradual decline of paganism and spread 


of Christianity, — these events were of far-reaching influence 
on the progress of the sciences, which then had their home in 
Alexandria. Alexandria became a commercial and intellectual 
emporium. Traders of all nations met in her busy streets, and 
in her magnificent Library, museums, lecture-halls, scholars 
from the East mingled with those of the West; Greeks began to 
study older literatures and to compare them with their own. In 
consequence of this interchange of ideas the Greek philosophy 
became fused with Oriental philosophy. Neo-Pythagoreanism 
and Neo-Platonism were the names of the modified systems. 
These stood, for a time, in opposition to Christianity. The 
study of Platonism and Pythagorean mysticism led to the 
revival of the theory of numbers. Perhaps the dispersion of 
the Jews and their introduction to Greek learning helped in 
bringing about this revival. The theory of numbers became 
a favourite study. This new line of mathematical inquiry 
ushered in what we may call a new school. There is no doubt 
that even now geometry continued to be one of the most 
important studies in the Alexandrian course. This Second 
Alexandrian School may be said to begin with the Christian 
era. It was made famous by the names of Claudius Ptolemaeus, 
Diophantus, Pappus, Theon of Smyrna, Theon of Alexandria, 
Iamblichus, Porphyrius, and others. 

By the side of these we may place Serenus of Antissa, 
as having been connected more or less with this new school. 
He wrote on sections of the cone and cylinder, in two books, 
one of which treated only of the triangular section of the 
cone through the apex. He solved the problem, "given 


a cone (cylinder), to find a cylinder (cone), so that the 
section of both by the same plane gives similar ellipses." 
Of particular interest is the following theorem, which is the 
foundation of the modern theory of harmonics: If from D we 
draw DF, cutting the tri- 
angle ABC, and choose ^ 
H on it, so that DE : 
DF = EH : HF, and if 
we draw the line AH, then 
every transversal through B ' 
D, such as DG, will be di- 
vided by AH so that DK : DG = KJ : JG. Menelaus of 
Alexandria (about 98 A.D.) was the author of Sphcerica, a 
work extant in Hebrew and Arabic, but not in Greek. In it he 
proves the theorems on the congruence of spherical triangles, 
and describes their properties in much the same way as Euclid 
treats plane triangles. In it are also found the theorems that 
the sum of the three sides of a spherical triangle is less than 
a great circle, and that the sum of the three angles exceeds 
two right angles. Celebrated are two theorems of his on plane 
and spherical triangles. The one on plane triangles is that, "if 
the three sides be cut by a straight line, the product of the 
three segments which have no common extremity is equal to 
the product of the other three." The illustrious Carnot makes 
this proposition, known as the 'lemma of Menelaus,' the base 
of his theory of transversals. The corresponding theorem for 
spherical triangles, the so-called 'regula sex quantitatum,' is 
obtained from the above by reading "chords of three segments 


doubled," in place of "three segments." 

Claudius Ptolemaeus, a celebrated astronomer, was a 
native of Egypt. Nothing is known of his personal history 
except that he flourished in Alexandria in 139 A.D. and that 
he made the earliest astronomical observations recorded in 
his works, in 125 A.D., the latest in 151 A.D. The chief of his 
works are the Syntaxis Mathematica (or the Almagest, as the 
Arabs call it) and the Geographica, both of which are extant. 
The former work is based partly on his own researches, but 
mainly on those of Hipparchus. Ptolemy seems to have been 
not so much of an independent investigator, as a corrector 
and improver of the work of his great predecessors. The 
Almagest forms the foundation of all astronomical science 
down to Copernicus. The fundamental idea of his system, 
the "Ptolemaic System," is that the earth is in the centre of 
the universe, and that the sun and planets revolve around the 
earth. Ptolemy did considerable for mathematics. He created, 
for astronomical use, a trigonometry remarkably perfect in 
form. The foundation of this science was laid by the illustrious 

The Almagest is in 13 books. Chapter 9 of the first book 
shows how to calculate tables of chords. The circle is divided 
into 360 degrees, each of which is halved. The diameter is 
divided into 120 divisions; each of these into 60 parts, which are 
again subdivided into 60 smaller parts. In Latin, these parts 
were called partes minutcE primas and partes minutce secundce. 
Hence our names, 'minutes' and 'seconds.' [3] The sexagesimal 
method of dividing the circle is of Babylonian origin, and was 


known to Geminus and Hipparchus. But Ptolemy's method 
of calculating chords seems original with him. He first proved 
the proposition, now appended to Euclid VI. (D), that "the 
rectangle contained by the diagonals of a quadrilateral figure 
inscribed in a circle is equal to both the rectangles contained 
by its opposite sides." He then shows how to find from the 
chords of two arcs the chords of their sum and difference, and 
from the chord of any arc that of its half. These theorems he 
applied to the calculation of his tables of chords. The proofs 
of these theorems are very pretty. 

Another chapter of the first book in the Almagest is 
devoted to trigonometry, and to spherical trigonometry in 
particular. Ptolemy proved the 'lemma of Menelaus,' and 
also the 'regula sex quantitatum.' Upon these propositions 
he built up his trigonometry. The fundamental theorem of 
plane trigonometry, that two sides of a triangle are to each 
other as the chords of double the arcs measuring the angles 
opposite the two sides, was not stated explicitly by him, but 
was contained implicitly in other theorems. More complete 
are the propositions in spherical trigonometry. 

The fact that trigonometry was cultivated not for its own 
sake, but to aid astronomical inquiry, explains the rather 
startling fact that spherical trigonometry came to exist in a 
developed state earlier than plane trigonometry. 

The remaining books of the Almagest are on astronomy. 
Ptolemy has written other works which have little or no 
bearing on mathematics, except one on geometry. Extracts 
from this book, made by Proclus, indicate that Ptolemy did 


not regard the parallel- axiom of Euclid as self-evident, and 
that Ptolemy was the first of the long line of geometers from 
ancient time down to our own who toiled in the vain attempt 
to prove it. 

Two prominent mathematicians of this time were Nico- 
machus and Theon of Smyrna. Their favourite study was 
theory of numbers. The investigations in this science culmi- 
nated later in the algebra of Diophantus. But no important 
geometer appeared after Ptolemy for 150 years. The only 
occupant of this long gap was Sextus Julius Africanus, 
who wrote an unimportant work on geometry applied to the 
art of war, entitled Cestes. 

Pappus, probably born about 340 A.D., in Alexandria, was 
the last great mathematician of the Alexandrian school. His 
genius was inferior to that of Archimedes, Apollonius, and 
Euclid, who flourished over 500 years earlier. But living, as he 
did, at a period when interest in geometry was declining, he 
towered above his contemporaries "like the peak of Teneriffa 
above the Atlantic." He is the author of a Commentary 
on the Almagest, a Commentary on Euclid's Elements, a 
Commentary on the Analemma of Diodorus, — a writer of 
whom nothing is known. All these works are lost. Proclus, 
probably quoting from the Commentary on Euclid, says that 
Pappus objected to the statement that an angle equal to a 
right angle is always itself a right angle. 

The only work of Pappus still extant is his Mathematical 
Collections. This was originally in eight books, but the first 
and portions of the second are now missing. The Mathematical 


Collections seems to have been written by Pappus to supply 
the geometers of his time with a succinct analysis of the most 
difficult mathematical works and to facilitate the study of 
them by explanatory lemmas. But these lemmas are selected 
very freely, and frequently have little or no connection with the 
subject on hand. However, he gives very accurate summaries 
of the works of which he treats. The Mathematical Collections 
is invaluable to us on account of the rich information it gives 
on various treatises by the foremost Greek mathematicians, 
which are now lost. Mathematicians of the last century 
considered it possible to restore lost works from the resume 
by Pappus alone. 

We shall now cite the more important of those theorems 
in the Mathematical Collections which are supposed to be 
original with Pappus. First of all ranks the elegant theorem 
re-discovered by Guldin, over 1000 years later, that the volume 
generated by the revolution of a plane curve which lies wholly 
on one side of the axis, equals the area of the curve multiplied 
by the circumference described by its centre of gravity. Pappus 
proved also that the centre of gravity of a triangle is that of 
another triangle whose vertices lie upon the sides of the first 
and divide its three sides in the same ratio. In the fourth book 
are new and brilliant propositions on the quadratrix which 
indicate an intimate acquaintance with curved surfaces. He 
generates the quadratrix as follows: Let a spiral line be drawn 
upon a right circular cylinder; then the perpendiculars to the 
axis of the cylinder drawn from each point of the spiral line 
form the surface of a screw. A plane passed through one of 


these perpendiculars, making any convenient angle with the 
base of the cylinder, cuts the screw-surface in a curve, the 
orthogonal projection of which upon the base is the quadratrix. 
A second mode of generation is no less admirable: If we make 
the spiral of Archimedes the base of a right cylinder, and 
imagine a cone of revolution having for its axis the side of the 
cylinder passing through the initial point of the spiral, then 
this cone cuts the cylinder in a curve of double curvature. 
The perpendiculars to the axis drawn through every point in 
this curve form the surface of a screw which Pappus here calls 
the plectoidal surface. A plane passed through one of the 
perpendiculars at any convenient angle cuts that surface in a 
curve whose orthogonal projection upon the plane of the spiral 
is the required quadratrix. Pappus considers curves of double 
curvature still further. He produces a spherical spiral by a 
point moving uniformly along the circumference of a great 
circle of a sphere, while the great circle itself revolves uniformly 
around its diameter. He then finds the area of that portion of 
the surface of the sphere determined by the spherical spiral, 
"a complanation which claims the more lively admiration, if 
we consider that, although the entire surface of the sphere was 
known since Archimedes' time, to measure portions thereof, 
such as spherical triangles, was then and for a long time 
afterwards an unsolved problem." [3] A question which was 
brought into prominence by Descartes and Newton is the 
"problem of Pappus." Given several straight lines in a plane, 
to find the locus of a point such that when perpendiculars 
(or, more generally, straight lines at given angles) are drawn 


from it to the given lines, the product of certain ones of them 
shall be in a given ratio to the product of the remaining 
ones. It is worth noticing that it was Pappus who first found 
the focus of the parabola, suggested the use of the directrix, 
and propounded the theory of the involution of points. He 
solved the problem to draw through three points lying in the 
same straight line, three straight lines which shall form a 
triangle inscribed in a given circle. [3] From the Mathematical 
Collections many more equally difficult theorems might be 
quoted which are original with Pappus as far as we know. It 
ought to be remarked, however, that he is known in three 
instances to have copied theorems without giving due credit, 
and that he may have done the same thing in other cases 
in which we have no data by which to ascertain the real 

About the time of Pappus lived Theon of Alexandria. He 
brought out an edition of Euclid's Elements with notes, which 
he probably used as a text-book in his classes. His commentary 
on the Almagest is valuable for the many historical notices, 
and especially for the specimens of Greek arithmetic which it 
contains. Theon's daughter Hypatia, a woman celebrated 
for her beauty and modesty, was the last Alexandrian teacher 
of reputation, and is said to have been an abler philosopher 
and mathematician than her father. Her notes on the works of 
Diophantus and Apollonius have been lost. Her tragic death 
in 415 A.D. is vividly described in Kingsley's Hypatia. 

From now on, mathematics ceased to be cultivated in 
Alexandria. The leading subject of men's thoughts was 


Christian theology. Paganism disappeared, and with it pagan 
learning. The Neo-Platonic school at Athens struggled on 
a century longer. Proclus, Isidorus, and others kept up the 
"golden chain of Platonic succession." Proclus, the successor 
of Syrianus, at the Athenian school, wrote a commentary on 
Euclid's Elements. We possess only that on the first book, 
which is valuable for the information it contains on the history 
of geometry. Damascius of Damascus, the pupil of Isidorus, 
is now believed to be the author of the fifteenth book of 
Euclid. Another pupil of Isidorus was Eutocius of Ascalon, 
the commentator of Apollonius and Archimedes. Simplicius 
wrote a commentary on Aristotle's De Coelo. In the year 529, 
Justinian, disapproving heathen learning, finally closed by 
imperial edict the schools at Athens. 

As a rule, the geometers of the last 500 years showed a 
lack of creative power. They were commentators rather than 

The principal characteristics of ancient geometry are:— 

(1) A wonderful clearness and definiteness of its concepts 
and an almost perfect logical rigour of its conclusions. 

(2) A complete want of general principles and methods. 
Ancient geometry is decidedly special. Thus the Greeks 
possessed no general method of drawing tangents. "The 
determination of the tangents to the three conic sections did 
not furnish any rational assistance for drawing the tangent 
to any other new curve, such as the conchoid, the cissoid, 
etc." [15] In the demonstration of a theorem, there were, 
for the ancient geometers, as many different cases requiring 


separate proof as there were different positions for the lines. 
The greatest geometers considered it necessary to treat all 
possible cases independently of each other, and to prove each 
with equal fulness. To devise methods by which the various 
cases could all be disposed of by one stroke, was beyond 
the power of the ancients. "If we compare a mathematical 
problem with a huge rock, into the interior of which we desire 
to penetrate, then the work of the Greek mathematicians 
appears to us like that of a vigorous stonecutter who, with 
chisel and hammer, begins with indefatigable perseverance, 
from without, to crumble the rock slowly into fragments; the 
modern mathematician appears like an excellent miner, who 
first bores through the rock some few passages, from which he 
then bursts it into pieces with one powerful blast, and brings 
to light the treasures within." [16] 


Greek mathematicians were in the habit of discriminating 
between the science of numbers and the art of calculation. 
The former they called arithmetica, the latter logistica. The 
drawing of this distinction between the two was very natural 
and proper. The difference between them is as marked as that 
between theory and practice. Among the Sophists the art of 
calculation was a favourite study. Plato, on the other hand, 
gave considerable attention to philosophical arithmetic, but 
pronounced calculation a vulgar and childish art. 

In sketching the history of Greek calculation, we shall first 


give a brief account of the Greek mode of counting and of 
writing numbers. Like the Egyptians and Eastern nations, 
the earliest Greeks counted on their fingers or with pebbles. 
In case of large numbers, the pebbles were probably arranged 
in parallel vertical lines. Pebbles on the first line represented 
units, those on the second tens, those on the third hundreds, 
and so on. Later, frames came into use, in which strings 
or wires took the place of lines. According to tradition, 
Pythagoras, who travelled in Egypt and, perhaps, in India, 
first introduced this valuable instrument into Greece. The 
abacus, as it is called, existed among different peoples and 
at different times, in various stages of perfection. An abacus 
is still employed by the Chinese under the name of Swan- 
pan. We possess no specific information as to how the Greek 
abacus looked or how it was used. Boethius says that the 
Pythagoreans used with the abacus certain nine signs called 
apices, which resembled in form the nine "Arabic numerals." 
But the correctness of this assertion is subject to grave doubts. 
The oldest Grecian numerical symbols were the so-called 
Herodianic signs (after Herodianus, a Byzantine grammarian 
of about 200 A.D., who describes them). These signs occur 
frequently in Athenian inscriptions and are, on that account, 
now generally called Attic. For some unknown reason these 
symbols were afterwards replaced by the alphabetic numerals, 
in which the letters of the Greek alphabet were used, together 
with three strange and antique letters 9, ? , and 7D , and the 
symbol M. This change was decidedly for the worse, for 
the old Attic numerals were less burdensome on the memory, 


inasmuch as they contained fewer symbols and were better 
adapted to show forth analogies in numerical operations. The 
following table shows the Greek alphabetic numerals and their 
respective values:— 

a (3 7 5 e 9 C V @ L KA/iz^£o7r9 
1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 

p a t v 4> x ^ w *9 , a fi ,1 e t c - 
100 200 300 400 500 600 700 800 900 1000 2000 3000 

P 7 

M M M etc. 

10,000 20,000 30,000 

It will be noticed that at 1000, the alphabet is begun over 
again, but, to prevent confusion, a stroke is now placed before 
the letter and generally somewhat below it. A horizontal line 
drawn over a number served to distinguish it more readily 
from words. The coefficient for M was sometimes placed 
before or behind instead of over the M. Thus 43,678 was 

written SMjxoV- It is to be observed that the Greeks had no 

Fractions were denoted by first writing the numerator 
marked with an accent, then the denominator marked with 
two accents and written twice. Thus, lj'k6"k0" = j|. In 
case of fractions having unity for the numerator, the a' was 
omitted and the denominator was written only once. Thus 
^" = ^. 

Greek writers seldom refer to calculation with alphabetic nu- 
merals. Addition, subtraction, and even multiplication were 
probably performed on the abacus. Expert mathematicians 




2 65 


40000, 12000, 


M fi rix t 

12000, 3600, 


fit T Kl 

1000, 300, 



M oTse 70225 


may have used the symbols. Thus Eutocius, a commen- 
tator of the sixth century after Christ, gives a great many 
multiplications of which the following is a specimen: [6]- 

The operation is explained 
sufficiently by the modern 
numerals appended. In case 
of mixed numbers, the pro- 
cess was still more clumsy. 
Divisions are found in Theon 
of Alexandria's commentary 
on the Almagest. As might 
be expected, the process is long and tedious. 

We have seen in geometry that the more advanced mathe- 
maticians frequently had occasion to extract the square root. 
Thus Archimedes in his Mensuration of the Circle gives a 
large number of square roots. He states, for instance, that 
a/3 < i^ and a/3 > f§§ , but he gives no clue to the method by 
which he obtained these approximations. It is not improbable 
that the earlier Greek mathematicians found the square root 
by trial only. Eutocius says that the method of extracting it 
was given by Heron, Pappus, Theon, and other commentators 
on the Almagest. Theon's is the only ancient method known 
to us. It is the same as the one used nowadays, except that 
sexagesimal fractions are employed in place of our decimals. 
What the mode of procedure actually was when sexagesimal 
fractions were not used, has been the subject of conjecture on 
the part of numerous modern writers. [17] 

Of interest, in connection with arithmetical symbolism, 


is the Sand-Counter (Arenarius), an essay addressed by 
Archimedes to Gelon, king of Syracuse. In it Archimedes 
shows that people are in error who think the sand cannot be 
counted, or that if it can be counted, the number cannot be 
expressed by arithmetical symbols. He shows that the number 
of grains in a heap of sand not only as large as the whole 
earth, but as large as the entire universe, can be arithmetically 
expressed. Assuming that 10, 000 grains of sand suffice to 
make a little solid of the magnitude of a poppy- seed, and that 
the diameter of a poppy-seed be not smaller than ^ part 
of a finger's breadth; assuming further, that the diameter of 
the universe (supposed to extend to the sun) be less than 
10, 000 diameters of the earth, and that the latter be less than 
1,000,000 stadia, Archimedes finds a number which would 
exceed the number of grains of sand in the sphere of the 
universe. He goes on even further. Supposing the universe to 
reach out to the fixed stars, he finds that the sphere, having 
the distance from the earth's centre to the fixed stars for its 
radius, would contain a number of grains of sand less than 
1000 myriads of the eighth octad. In our notation, this number 
would be 10 63 or 1 with 63 ciphers after it. It can hardly 
be doubted that one object which Archimedes had in view 
in making this calculation was the improvement of the Greek 
symbolism. It is not known whether he invented some short 
notation by which to represent the above number or not. 

We judge from fragments in the second book of Pappus that 
Apollonius proposed an improvement in the Greek method 
of writing numbers, but its nature we do not know. Thus 


we see that the Greeks never possessed the boon of a clear, 
comprehensive symbolism. The honour of giving such to the 
world, once for all, was reserved by the irony of fate for a 
nameless Indian of an unknown time, and we know not whom 
to thank for an invention of such importance to the general 
progress of intelligence. [6] 

Passing from the subject of logistica to that of arithmetica, 
our attention is first drawn to the science of numbers of 
Pythagoras. Before founding his school, Pythagoras studied 
for many years under the Egyptian priests and familiarised 
himself with Egyptian mathematics and mysticism. If he ever 
was in Babylon, as some authorities claim, he may have learned 
the sexagesimal notation in use there; he may have picked 
up considerable knowledge on the theory of proportion, and 
may have found a large number of interesting astronomical 
observations. Saturated with that speculative spirit then 
pervading the Greek mind, he endeavoured to discover some 
principle of homogeneity in the universe. Before him, the 
philosophers of the Ionic school had sought it in the matter 
of things; Pythagoras looked for it in the structure of things. 
He observed various numerical relations or analogies between 
numbers and the phenomena of the universe. Being convinced 
that it was in numbers and their relations that he was to 
find the foundation to true philosophy, he proceeded to trace 
the origin of all things to numbers. Thus he observed that 
musical strings of equal length stretched by weights having 
the proportion of %, |, |, produced intervals which were an 
octave, a fifth, and a fourth. Harmony, therefore, depends on 


musical proportion; it is nothing but a mysterious numerical 
relation. Where harmony is, there are numbers. Hence the 
order and beauty of the universe have their origin in numbers. 
There are seven intervals in the musical scale, and also seven 
planets crossing the heavens. The same numerical relations 
which underlie the former must underlie the latter. But 
where numbers are, there is harmony. Hence his spiritual ear 
discerned in the planetary motions a wonderful 'harmony of 
the spheres.' The Pythagoreans invested particular numbers 
with extraordinary attributes. Thus one is the essence of 
things; it is an absolute number; hence the origin of all numbers 
and so of all things. Four is the most perfect number, and was 
in some mystic way conceived to correspond to the human 
soul. Philolaus believed that 5 is the cause of color, 6 of cold, 
7 of mind and health and light, 8 of love and friendship. [6] 
In Plato's works are evidences of a similar belief in religious 
relations of numbers. Even Aristotle referred the virtues to 

Enough has been said about these mystic speculations to 
show what lively interest in mathematics they must have 
created and maintained. Avenues of mathematical inquiry 
were opened up by them which otherwise would probably 
have remained closed at that time. 

The Pythagoreans classified numbers into odd and even. 
They observed that the sum of the series of odd numbers from 
1 to 2n + 1 was always a complete square, and that by addition 
of the even numbers arises the series 2, 6, 12, 20, in which 
every number can be decomposed into two factors differing 


from each other by unity. Thus, 6 = 2-3, 12 = 3-4, etc. These 
latter numbers were considered of sufficient importance to 
receive the separate name of heteromecic (not equilateral). [7] 

Numbers of the form were called triangular, because 

2 • 

they could always be arranged thus, •*•*• • Numbers which 

were equal to the sum of all their possible factors, such as 6, 28, 

496, were called perfect; those exceeding that sum, excessive; 

and those which were less, defective. Amicable numbers were 

those of which each was the sum of the factors in the other. 

Much attention was paid by the Pythagoreans to the subject 

of proportion. The quantities a, 6, c, d were said to be in 

arithmetical proportion when a — b = c — d; in geometrical 

proportion, when a : b = c : d; in harmonic proportion, when 

a — b : b — c = a : c. It is probable that the Pythagoreans were 

also familiar with the musical proportion a : = : b. 

F F 2 a+b 

Iamblichus says that Pythagoras introduced it from Babylon. 

In connection with arithmetic, Pythagoras made extensive 
investigations into geometry. He believed that an arithmetical 
fact had its analogue in geometry, and vice versa. In connection 
with his theorem on the right triangle he devised a rule by 
which integral numbers could be found, such that the sum 
of the squares of two of them equalled the square of the 

third. Thus, take for one side an odd number (2n + 1); then 

(2n + 1) 2 - 1 9 . . . . , , 9 

•i - = 2n 2 + 2n = the other side, and (2n l + 2n + 1) = 

hypotenuse. If 2n + 1 = 9, then the other two numbers are 

40 and 41. But this rule only applies to cases in which the 

hypotenuse differs from one of the sides by 1. In the study of 


the right triangle there doubtless arose questions of puzzling 
subtlety. Thus, given a number equal to the side of an isosceles 
right triangle, to find the number which the hypotenuse is 
equal to. The side may have been taken equal to 1, 2, |, 
|, or any other number, yet in every instance all efforts to 
find a number exactly equal to the hypotenuse must have 
remained fruitless. The problem may have been attacked 
again and again, until finally "some rare genius, to whom it 
is granted, during some happy moments, to soar with eagle's 
flight above the level of human thinking," grasped the happy 
thought that this problem cannot be solved. In some such 
manner probably arose the theory of irrational quantities, 
which is attributed by Eudemus to the Pythagoreans. It 
was indeed a thought of extraordinary boldness, to assume 
that straight lines could exist, differing from one another 
not only in length, — that is, in quantity, — but also in a 
quality, which, though real, was absolutely invisible. [7] Need 
we wonder that the Pythagoreans saw in irrationals a deep 
mystery, a symbol of the unspeakable? We are told that 
the one who first divulged the theory of irrationals, which 
the Pythagoreans kept secret, perished in consequence in a 
shipwreck. Its discovery is ascribed to Pythagoras, but we 
must remember that all important Pythagorean discoveries 
were, according to Pythagorean custom, referred back to 
him. The first incommensurable ratio known seems to have 
been that of the side of a square to its diagonal, as 1 : \[2. 
Theodorus of Cyrene added to this the fact that the sides 
of squares represented in length by a/3, v 7 ^, etc., up to vT7, 


and Theaetetus, that the sides of any square, represented by 
a surd, are incommensurable with the linear unit. Euclid 
(about 300 B.C.), in his Elements, X. 9, generalised still 
further: Two magnitudes whose squares are (or are not) 
to one another as a square number to a square number are 
commensurable (or incommensurable), and conversely. In 
the tenth book, he treats of incommensurable quantities at 
length. He investigates every possible variety of lines which 

can be represented by y Va ± Vb, a and b representing two 
commensurable lines, and obtains 25 species. Every individual 
of every species is incommensurable with all the individuals 
of every other species. "This book," says De Morgan, "has 
a completeness which none of the others (not even the fifth) 
can boast of; and we could almost suspect that Euclid, having 
arranged his materials in his own mind, and having completely 
elaborated the tenth book, wrote the preceding books after it, 
and did not live to revise them thoroughly." [9] The theory 
of incommensurables remained where Euclid left it, till the 
fifteenth century. 

Euclid devotes the seventh, eighth, and ninth books of his 
Elements to arithmetic. Exactly how much contained in these 
books is Euclid's own invention, and how much is borrowed 
from his predecessors, we have no means of knowing. Without 
doubt, much is original with Euclid. The seventh book begins 
with twenty-one definitions. All except that for 'prime' 
numbers are known to have been given by the Pythagoreans. 
Next follows a process for finding the G.C.D. of two or more 
numbers. The eighth book deals with numbers in continued 


proportion, and with the mutual relations of squares, cubes, 
and plane numbers. Thus, XXII. , if three numbers are in 
continued proportion, and the first is a square, so is the third. 
In the ninth book, the same subject is continued. It contains 
the proposition that the number of primes is greater than any 
given number. 

After the death of Euclid, the theory of numbers remained 
almost stationary for 400 years. Geometry monopolised 
the attention of all Greek mathematicians. Only two are 
known to have done work in arithmetic worthy of mention. 
Eratosthenes (275-194 B.C.) invented a 'sieve' for finding 
prime numbers. All composite numbers are 'sifted' out in 
the following manner: Write down the odd numbers from 
3 up, in succession. By striking out every third number after 
the 3, we remove all multiples of 3. By striking out every 
fifth number after the 5, we remove all multiples of 5. In 
this way, by rejecting multiples of 7, 11, 13, etc., we have left 
prime numbers only. Hypsicles (between 200 and 100 B.C.) 
worked at the subjects of polygonal numbers and arithmetical 
progressions, which Euclid entirely neglected. In his work on 
'risings of the stars,' he showed (1) that in an arithmetical 
series of In terms, the sum of the last n terms exceeds the 
sum of the first n by a multiple of n 2 ; (2) that in such a series 
of 2n + 1 terms, the sum of the series is the number of terms 
multiplied by the middle term; (3) that in such a series of 
2n terms, the sum is half the number of terms multiplied by 
the two middle terms. [6] 

For two centuries after the time of Hypsicles, arithmetic 


disappears from history. It is brought to light again about 
100 A.D. by Nicomachus, a Neo-Pythagorean, who inau- 
gurated the final era of Greek mathematics. From now 
on, arithmetic was a favourite study, while geometry was 
neglected. Nicomachus wrote a work entitled Introductio 
Arithmetica, which was very famous in its day. The great 
number of commentators it has received vouch for its popu- 
larity. Boethius translated it into Latin. Lucian could pay no 
higher compliment to a calculator than this: "You reckon like 
Nicomachus of Gerasa." The Introductio Arithmetica was the 
first exhaustive work in which arithmetic was treated quite 
independently of geometry. Instead of drawing lines, like 
Euclid, he illustrates things by real numbers. To be sure, in 
his book the old geometrical nomenclature is retained, but the 
method is inductive instead of deductive. "Its sole business is 
classification, and all its classes are derived from, and exhibited 
by, actual numbers." The work contains few results that are 
really original. We mention one important proposition which 
is probably the author's own. He states that cubical numbers 
are always equal to the sum of successive odd numbers. Thus, 
8 = 2 3 = 3 + 5, 27 = 3 3 = 7 + 9 + 11,64 = 4 3 = 13 + 15 + 17+19, 
and so on. This theorem was used later for finding the sum 
of the cubical numbers themselves. Theon of Smyrna is the 
author of a treatise on "the mathematical rules necessary for 
the study of Plato." The work is ill arranged and of little 
merit. Of interest is the theorem, that every square number, 
or that number minus 1, is divisible by 3 or 4 or both. A 
remarkable discovery is a proposition given by Iamblichus 


in his treatise on Pythagorean philosophy. It is founded on 
the observation that the Pythagoreans called 1, 10, 100, 1000, 
units of the first, second, third, fourth 'course' respectively. 
The theorem is this: If we add any three consecutive numbers, 
of which the highest is divisible by 3, then add the digits of 
that sum, then, again, the digits of that sum, and so on, the 
final sum will be 6. Thus, 61 + 62 + 63 = 186, 1 + 8 + 6 = 15, 
1 + 5 = 6. This discovery was the more remarkable, because 
the ordinary Greek numerical symbolism was much less likely 
to suggest any such property of numbers than our "Arabic" 
notation would have been. 

The works of Nicomachus, Theon of Smyrna, Thymaridas, 
and others contain at times investigations of subjects which are 
really algebraic in their nature. Thymaridas in one place uses 
the Greek word meaning "unknown quantity" in a way which 
would lead one to believe that algebra was not far distant. Of 
interest in tracing the invention of algebra are the arithmetical 
epigrams in the Palatine Anthology, which contain about fifty 
problems leading to linear equations. Before the introduction 
of algebra these problems were propounded as puzzles. A 
riddle attributed to Euclid and contained in the Anthology is 
to this effect: A mule and a donkey were walking along, laden 
with corn. The mule says to the donkey, "If you gave me 
one measure, I should carry twice as much as you. If I gave 
you one, we should both carry equal burdens. Tell me their 
burdens, O most learned master of geometry." [6] 

It will be allowed, says Gow, that this problem, if authentic, 
was not beyond Euclid, and the appeal to geometry smacks of 


antiquity. A far more difficult puzzle was the famous 'cattle- 
problem,' which Archimedes propounded to the Alexandrian 
mathematicians. The problem is indeterminate, for from 
only seven equations, eight unknown quantities in integral 
numbers are to be found. It may be stated thus: The sun had 
a herd of bulls and cows, of different colours. (1) Of Bulls, 
the white (W) were, in number, (^ + 3) of the blue (B) and 
yellow (Y): the B were (| + |) of the Y and piebald (P): the 
P were ( jj + 7) of the W and Y. (2) Of Cows, which had the 
same colours (w, b, y, p), 

w = {\ + \){B + b) :b= {\+\){P +p) : V 

= (i + l)(Y + y) --y = (i + i)(w + w). 

Find the number of bulls and cows. [6] Another problem in the 
Anthology is quite familiar to school-boys: "Of four pipes, one 
fills the cistern in one day, the next in two days, the third in 
three days, the fourth in four days: if all run together, how soon 
will they fill the cistern?" A great many of these problems, 
puzzling to an arithmetician, would have been solved easily 
by an algebraist. They became very popular about the time 
of Diophantus, and doubtless acted as a powerful stimulus on 
his mind. 

Diophantus was one of the last and most fertile mathe- 
maticians of the second Alexandrian school. He died about 
330 A.D. His age was eighty- four, as is known from an epitaph 
to this effect: Diophantus passed g of his life in childhood, 
Y2 in youth, and i more as a bachelor; five years after his 
marriage was born a son who died four years before his father, 


at half his father's age. The place of nativity and parentage 
of Diophantus are unknown. If his works were not written in 
Greek, no one would think for a moment that they were the 
product of Greek mind. There is nothing in his works that 
reminds us of the classic period of Greek mathematics. His 
were almost entirely new ideas on a new subject. In the circle 
of Greek mathematicians he stands alone in his specialty. 
Except for him, we should be constrained to say that among 
the Greeks algebra was always an unknown science. 

Of his works we have lost the Porisms, but possess a 
fragment of Polygonal Numbers, and seven books of his great 
work on Arithmetica, said to have been written in 13 books. 

If we except the Ahmes papyrus, which contains the first 
suggestions of algebraic notation, and of the solution of 
equations, then his Arithmetica is the earliest treatise on 
algebra now extant. In this work is introduced the idea of 
an algebraic equation expressed in algebraic symbols. His 
treatment is purely analytical and completely divorced from 
geometrical methods. He is, as far as we know, the first 
to state that "a negative number multiplied by a negative 
number gives a positive number." This is applied to the 
multiplication of differences, such as (x — l)(x — 2). It must be 
remarked, however, that Diophantus had no notion whatever 
of negative numbers standing by themselves. All he knew 
were differences, such as (2x — 10), in which 2x could not 
be smaller than 10 without leading to an absurdity. He 
appears to be the first who could perform such operations as 
(x-l)x(x — 2) without reference to geometry. Such identities 


as (a + ft) 2 = a 2 + 2aft + ft 2 , which with Euclid appear in the 
elevated rank of geometric theorems, are with Diophantus 
the simplest consequences of the algebraic laws of operation. 
His sign for subtraction was W, for equality l. For unknown 
quantities he had only one symbol, q. He had no sign for 
addition except juxtaposition. Diophantus used but few 
symbols, and sometimes ignored even these by describing an 
operation in words when the symbol would have answered 
just as well. 

In the solution of simultaneous equations Diophantus 
adroitly managed with only one symbol for the unknown 
quantities and arrived at answers, most commonly, by the 
method of tentative assumption, which consists in assigning 
to some of the unknown quantities preliminary values, that 
satisfy only one or two of the conditions. These values lead 
to expressions palpably wrong, but which generally suggest 
some stratagem by which values can be secured satisfying all 
the conditions of the problem. 

Diophantus also solved determinate equations of the second 
degree. We are ignorant of his method, for he nowhere goes 
through with the whole process of solution, but merely states 
the result. Thus, "84x 2 + 7x = 7, whence x is found = |." 
Notice he gives only one root. His failure to observe that 
a quadratic equation has two roots, even when both roots 
are positive, rather surprises us. It must be remembered, 
however, that this same inability to perceive more than one 
out of the several solutions to which a problem may point 
is common to all Greek mathematicians. Another point to 


be observed is that he never accepts as an answer a quantity 
which is negative or irrational. 

Diophantus devotes only the first book of his Arithmetica to 
the solution of determinate equations. The remaining books 
extant treat mainly of indeterminate quadratic equations of 
the form Ax 2 + Bx + C = y 2 , or of two simultaneous equations 
of the same form. He considers several but not all the possible 
cases which may arise in these equations. The opinion of 
Nesselmann on the method of Diophantus, as stated by Gow, 
is as follows: "(1) Indeterminate equations of the second 
degree are treated completely only when the quadratic or 
the absolute term is wanting: his solution of the equations 
Ax 2 + C = y 2 and Ax 2 + Bx + C = y 2 is in many respects 
cramped. (2) For the 'double equation' of the second degree he 
has a definite rule only when the quadratic term is wanting in 
both expressions: even then his solution is not general. More 
complicated expressions occur only under specially favourable 
circumstances." Thus, he solves Bx + C 2 = y 2 , B\x + C 2 = y\. 

The extraordinary ability of Diophantus lies rather in 
another direction, namely, in his wonderful ingenuity to 
reduce all sorts of equations to particular forms which he 
knows how to solve. Very great is the variety of problems 
considered. The 130 problems found in the great work of 
Diophantus contain over 50 different classes of problems, which 
are strung together without any attempt at classification. But 
still more multifarious than the problems are the solutions. 
General methods are unknown to Diophantus. Each problem 
has its own distinct method, which is often useless for the 


most closely related problems. "It is, therefore, difficult for 
a modern, after studying 100 Diophantine solutions, to solve 
the 101st." [7] 

That which robs his work of much of its scientific value is the 
fact that he always feels satisfied with one solution, though 
his equation may admit of an indefinite number of values. 
Another great defect is the absence of general methods. 
Modern mathematicians, such as Euler, Lagrange, Gauss, 
had to begin the study of indeterminate analysis anew and 
received no direct aid from Diophantus in the formulation of 
methods. In spite of these defects we cannot fail to admire 
the work for the wonderful ingenuity exhibited therein in the 
solution of particular equations. 

It is still an open question and one of great difficulty 
whether Diophantus derived portions of his algebra from 
Hindoo sources or not. 


Nowhere is the contrast between the Greek and Roman 
mind shown forth more distinctly than in their attitude 
toward the mathematical science. The sway of the Greek was 
a flowering time for mathematics, but that of the Roman a 
period of sterility. In philosophy, poetry, and art the Roman 
was an imitator. But in mathematics he did not even rise to 
the desire for imitation. The mathematical fruits of Greek 
genius lay before him untasted. In him a science which had no 
direct bearing on practical life could awake no interest. As a 


consequence, not only the higher geometry of Archimedes and 
Apollonius, but even the Elements of Euclid, were entirely 
neglected. What little mathematics the Romans possessed 
did not come from the Greeks, but from more ancient sources. 
Exactly where and how it originated is a matter of doubt. 
It seems most probable that the "Roman notation," as well 
as the practical geometry of the Romans, came from the old 
Etruscans, who, at the earliest period to which our knowledge 
of them extends, inhabited the district between the Arno and 

Livy tells us that the Etruscans were in the habit of 
representing the number of years elapsed, by driving yearly 
a nail into the sanctuary of Minerva, and that the Romans 
continued this practice. A less primitive mode of designating 
numbers, presumably of Etruscan origin, was a notation 
resembling the present "Roman notation." This system is 
noteworthy from the fact that a principle is involved in it 
which is not met with in any other; namely, the principle of 
subtraction. If a letter be placed before another of greater 
value, its value is not to be added to, but subtracted from, 
that of the greater. In the designation of large numbers a 
horizontal bar placed over a letter was made to increase its 
value one thousand fold. In fractions the Romans used the 
duodecimal system. 

Of arithmetical calculations, the Romans employed three 
different kinds: Reckoning on the fingers, upon the abacus, 
and by tables prepared for the purpose. [3] Finger-symbolism 
was known as early as the time of King Numa, for he had 


erected, says Pliny, a statue of the double-faced Janus, of which 
the fingers indicated 365 (355?), the number of days in a year. 
Many other passages from Roman authors point out the use of 
the fingers as aids to calculation. In fact, a finger-symbolism 
of practically the same form was in use not only in Rome, but 
also in Greece and throughout the East, certainly as early as 
the beginning of the Christian era, and continued to be used 
in Europe during the Middle Ages. We possess no knowledge 
as to where or when it was invented. The second mode 
of calculation, by the abacus, was a subject of elementary 
instruction in Rome. Passages in Roman writers indicate that 
the kind of abacus most commonly used was covered with dust 
and then divided into columns by drawing straight lines. Each 
column was supplied with pebbles (calculi, whence 'calculare' 
and 'calculate') which served for calculation. Additions and 
subtractions could be performed on the abacus quite easily, 
but in multiplication the abacus could be used only for adding 
the particular products, and in division for performing the 
subtractions occurring in the process. Doubtless at this 
point recourse was made to mental operations and to the 
multiplication table. Possibly finger-multiplication may also 
have been used. But the multiplication of large numbers 
must, by either method, have been beyond the power of 
the ordinary arithmetician. To obviate this difficulty, the 
arithmetical tables mentioned above were used, from which 
the desired products could be copied at once. Tables of this 
kind were prepared by Victorius of Aquitania. His tables 
contain a peculiar notation for fractions, which continued in 


use throughout the Middle Ages. Victorius is best known 
for his canon paschalis, a rule for finding the correct date for 
Easter, which he published in 457 A.D. 

Payments of interest and problems in interest were very 
old among the Romans. The Roman laws of inheritance gave 
rise to numerous arithmetical examples. Especially unique is 
the following: A dying man wills that, if his wife, being with 
child, gives birth to a son, the son shall receive | and she j of 
his estates; but if a daughter is born, she shall receive ^ and 
his wife |. It happens that twins are born, a boy and a girl. 
How shall the estates be divided so as to satisfy the will? The 
celebrated Roman jurist, Salvianus Julianus, decided that the 
estates shall be divided into seven equal parts, of which the 
son receives four, the wife two, the daughter one. 

We next consider Roman geometry. He who expects 
to find in Rome a science of geometry, with definitions, 
axioms, theorems, and proofs arranged in logical order, will 
be disappointed. The only geometry known was a practical 
geometry, which, like the old Egyptian, consisted only of 
empirical rules. This practical geometry was employed in 
surveying. Treatises thereon have come down to us, compiled 
by the Roman surveyors, called agrimensores or gromatici. 
One would naturally expect rules to be clearly formulated. 
But no; they are left to be abstracted by the reader from a mass 
of numerical examples. "The total impression is as though 
the Roman gromatic were thousands of years older than 
Greek geometry, and as though a deluge were lying between 
the two." Some of their rules were probably inherited from 


the Etruscans, but others are identical with those of Heron. 
Among the latter is that for finding the area of a triangle from 
its sides and the approximate formula, gjj a 2 , for the area of 
equilateral triangles (a being one of the sides). But the latter 
area was also calculated by the formulas ^(a 2 + a) and \a 2 1 the 
first of which was unknown to Heron. Probably the expression 

T)d 2 was derived from the Egyptian formula for the 

determination of the surface of a quadrilateral. This Egyptian 
formula was used by the Romans for finding the area, not only 
of rectangles, but of any quadrilaterals whatever. Indeed, the 
gromatici considered it even sufficiently accurate to determine 
the areas of cities, laid out irregularly, simply by measuring 
their circumferences. [7] Whatever Egyptian geometry the 
Romans possessed was transplanted across the Mediterranean 
at the time of Julius Cdssar, who ordered a survey of the whole 
empire to secure an equitable mode of taxation. Caesar also 
reformed the calendar, and, for that purpose, drew from 
Egyptian learning. He secured the services of the Alexandrian 
astronomer, Sosigenes. 

In the fifth century, the Western Roman Empire was fast 
falling to pieces. Three great branches — Spain, Gaul, and the 
province of Africa — broke off from the decaying trunk. In 476, 
the Western Empire passed away, and the Visigothic chief, 
Odoacer, became king. Soon after, Italy was conquered by 
the Ostrogoths under Theodoric. It is remarkable that this 
very period of political humiliation should be the one during 
which Greek science was studied in Italy most zealously. 
School-books began to be compiled from the elements of 


Greek authors. These compilations are very deficient, but 
are of absorbing interest, from the fact that, down to the 
twelfth century, they were the only sources of mathematical 
knowledge in the Occident. Foremost among these writers 
is Boethius (died 524). At first he was a great favourite of 
King Theodoric, but later, being charged by envious courtiers 
with treason, he was imprisoned, and at last decapitated. 
While in prison he wrote On the Consolations of Philosophy. 
As a mathematician, Boethius was a Brobdingnagian among 
Roman scholars, but a Liliputian by the side of Greek masters. 
He wrote an Institutis Arithmetical which is essentially a 
translation of the arithmetic of Nicomachus, and a Geometry 
in several books. Some of the most beautiful results of 
Nicomachus are omitted in Boethius' arithmetic. The first 
book on geometry is an extract from Euclid's Elements, which 
contains, in addition to definitions, postulates, and axioms, 
the theorems in the first three books, without proofs. How 
can this omission of proofs be accounted for? It has been 
argued by some that Boethius possessed an incomplete Greek 
copy of the Elements; by others, that he had Theon's edition 
before him, and believed that only the theorems came from 
Euclid, while the proofs were supplied by Theon. The second 
book, as also other books on geometry attributed to Boethius, 
teaches, from numerical examples, the mensuration of plane 
figures after the fashion of the agrimensores. 

A celebrated portion in the geometry of Boethius is that 
pertaining to an abacus, which he attributes to the Pythago- 
reans. A considerable improvement on the old abacus is there 


introduced. Pebbles are discarded, and apices (probably 
small cones) are used. Upon each of these apices is drawn 
a numeral giving it some value below 10. The names of 
these numerals are pure Arabic, or nearly so, but are added, 
apparently, by a later hand. These figures are obviously the 
parents of our modern "Arabic" numerals. The is not men- 
tioned by Boethius in the text. These numerals bear striking 
resemblance to the Gubar-numerals of the West- Arabs, which 
are admittedly of Indian origin. These facts have given rise 
to an endless controversy. Some contended that Pythagoras 
was in India, and from there brought the nine numerals to 
Greece, where the Pythagoreans used them secretly. This 
hypothesis has been generally abandoned, for it is not certain 
that Pythagoras or any disciple of his ever was in India, nor is 
there any evidence in any Greek author, that the apices were 
known to the Greeks, or that numeral signs of any sort were 
used by them with the abacus. It is improbable, moreover, 
that the Indian signs, from which the apices are derived, are 
so old as the time of Pythagoras. A second theory is that the 
Geometry attributed to Boethius is a forgery; that it is not 
older than the tenth, or possibly the ninth, century, and that 
the apices are derived from the Arabs. This theory is based 
on contradictions between passages in the Arithmetica and 
others in the Geometry. But there is an Encyclopaedia written 
by Cassiodorius (died about 570) in which both the arithmetic 
and geometry of Boethius are mentioned. There appears to 
be no good reason for doubting the trustworthiness of this 
passage in the Encyclopaedia. A third theory (Woepcke's) is 


that the Alexandrians either directly or indirectly obtained 
the nine numerals from the Hindoos, about the second cen- 
tury a.d. , and gave them to the Romans on the one hand, and 
to the Western Arabs on the other. This explanation is the 
most plausible. 



The first people who distinguished themselves in math- 
ematical research, after the time of the ancient Greeks, 
belonged, like them, to the Aryan race. It was, however, not 
a European, but an Asiatic nation, and had its seat in far-off 

Unlike the Greek, Indian society was fixed into castes. The 
only castes enjoying the privilege and leisure for advanced 
study and thinking were the Brahmins, whose prime business 
was religion and philosophy, and the Kshatriyas, who attended 
to war and government. 

Of the development of Hindoo mathematics we know but 
little. A few manuscripts bear testimony that the Indians 
had climbed to a lofty height, but their path of ascent is 
no longer traceable. It would seem that Greek mathematics 
grew up under more favourable conditions than the Hindoo, 
for in Greece it attained an independent existence, and was 
studied for its own sake, while Hindoo mathematics always 
remained merely a servant to astronomy. Furthermore, in 
Greece mathematics was a science of the people, free to be 
cultivated by all who had a liking for it; in India, as in Egypt, it 
was in the hands chiefly of the priests. Again, the Indians were 
in the habit of putting into verse all mathematical results they 



obtained, and of clothing them in obscure and mystic language, 
which, though well adapted to aid the memory of him who 
already understood the subject, was often unintelligible to 
the uninitiated. Although the great Hindoo mathematicians 
doubtless reasoned out most or all of their discoveries, yet 
they were not in the habit of preserving the proofs, so that the 
naked theorems and processes of operation are all that have 
come down to our time. Very different in these respects were 
the Greeks. Obscurity of language was generally avoided, 
and proofs belonged to the stock of knowledge quite as much 
as the theorems themselves. Very striking was the difference 
in the bent of mind of the Hindoo and Greek; for, while 
the Greek mind was pre-eminently geometrical, the Indian 
was first of all arithmetical. The Hindoo dealt with number, 
the Greek with form. Numerical symbolism, the science of 
numbers, and algebra attained in India far greater perfection 
than they had previously reached in Greece. On the other 
hand, we believe that there was little or no geometry in India 
of which the source may not be traced back to Greece. Hindoo 
trigonometry might possibly be mentioned as an exception, 
but it rested on arithmetic more than on geometry. 

An interesting but difficult task is the tracing of the 
relation between Hindoo and Greek mathematics. It is 
well known that more or less trade was carried on between 
Greece and India from early times. After Egypt had become 
a Roman province, a more lively commercial intercourse 
sprang up between Rome and India, by way of Alexandria. 
A priori, it does not seem improbable, that with the traffic 


of merchandise there should also be an interchange of ideas. 
That communications of thought from the Hindoos to the 
Alexandrians actually did take place, is evident from the 
fact that certain philosophic and theologic teachings of the 
Manicheans, Neo-Platonists, Gnostics, show unmistakable 
likeness to Indian tenets. Scientific facts passed also from 
Alexandria to India. This is shown plainly by the Greek origin 
of some of the technical terms used by the Hindoos. Hindoo 
astronomy was influenced by Greek astronomy. Most of the 
geometrical knowledge which they possessed is traceable to 
Alexandria, and to the writings of Heron in particular. In 
algebra there was, probably, a mutual giving and receiving. 
We suspect that Diophantus got the first glimpses of algebraic 
knowledge from India. On the other hand, evidences have 
been found of Greek algebra among the Brahmins. The 
earliest knowledge of algebra in India may possibly have 
been of Babylonian origin. When we consider that Hindoo 
scientists looked upon arithmetic and algebra merely as tools 
useful in astronomical research, there appears deep irony in 
the fact that these secondary branches were after all the only 
ones in which they won real distinction, while in their pet 
science of astronomy they displayed an inaptitude to observe, 
to collect facts, and to make inductive investigations. 

We shall now proceed to enumerate the names of the 
leading Hindoo mathematicians, and then to review briefly 
Indian mathematics. We shall consider the science only in 
its complete state, for our data are not sufficient to trace the 
history of the development of methods. Of the great Indian 


mathematicians, or rather, astronomers, — for India had no 
mathematicians proper, — Aryabhatta is the earliest. He 
was born 476 A.D., at Pataliputra, on the upper Ganges. 
His celebrity rests on a work entitled Aryabhattiyam, of 
which the third chapter is devoted to mathematics. About 
one hundred years later, mathematics in India reached the 
highest mark. At that time flourished Brahmagupta 
(born 598). In 628 he wrote his Brahma-sphuta-siddhanta 
("The Revised System of Brahma"), of which the twelfth 
and eighteenth chapters belong to mathematics. To the 
fourth or fifth century belongs an anonymous astronomical 
work, called Surya-siddhanta ("Knowledge from the Sun"), 
which by native authorities was ranked second only to the 
Brahma-siddhanta, but is of interest to us merely as furnishing 
evidence that Greek science influenced Indian science even 
before the time of Aryabhatta. The following centuries 
produced only two names of importance; namely, Cridhara, 
who wrote a Ganita-sara ("Quintessence of Calculation"), 
and Padmanabha, the author of an algebra. The science 
seems to have made but little progress at this time; for a work 
entitled Siddhantaciromani ("Diadem of an Astronomical 
System"), written by Bhaskara Acarya in 1150, stands 
little higher than that of Brahmagupta, written over 500 years 
earlier. The two most important mathematical chapters in 
this work are the Lilavati (= "the beautiful," i.e. the noble 
science) and Viga-ganita (= "root-extraction"), devoted to 
arithmetic and algebra. From now on, the Hindoos in the 
Brahmin schools seemed to content themselves with studying 


the masterpieces of their predecessors. Scientific intelligence 
decreases continually, and in modern times a very deficient 
Arabic work of the sixteenth century has been held in great 
authority. [7] 

The mathematical chapters of the Brahma- siddhanta and 
Siddhantaciromani were translated into English by H. T. 
Colebrooke, London, 1817. The Sury a- siddhanta was trans- 
lated by E. Burgess, and annotated by W. D. Whitney, New 
Haven, Conn., 1860. 

The grandest achievement of the Hindoos and the one 
which, of all mathematical inventions, has contributed most 
to the general progress of intelligence, is the invention of the 
principle of position in writing numbers. Generally we speak 
of our notation as the "Arabic" notation, but it should be 
called the "Hindoo" notation, for the Arabs borrowed it from 
the Hindoos. That the invention of this notation was not so 
easy as we might suppose at first thought, may be inferred 
from the fact that, of other nations, not even the keen-minded 
Greeks possessed one like it. We inquire, who invented this 
ideal symbolism, and when? But we know neither the inventor 
nor the time of invention. That our system of notation is 
of Indian origin is the only point of which we are certain. 
From the evolution of ideas in general we may safely infer 
that our notation did not spring into existence a completely 
armed Minerva from the head of Jupiter. The nine figures 
for writing the units are supposed to have been introduced 
earliest, and the sign of zero and the principle of position to be 
of later origin. This view receives support from the fact that 


on the island of Ceylon a notation resembling the Hindoo, but 
without the zero has been preserved. We know that Buddhism 
and Indian culture were transplanted to Ceylon about the 
third century after Christ, and that this culture remained 
stationary there, while it made progress on the continent. It 
seems highly probable, then, that the numerals of Ceylon are 
the old, imperfect numerals of India. In Ceylon, nine figures 
were used for the units, nine others for the tens, one for 100, 
and also one for 1000. These 20 characters enabled them to 
write all the numbers up to 9999. Thus, 8725 would have been 
written with six signs, representing the following numbers: 
8, 1000, 7, 100, 20, 5. These Singhalesian signs, like the old 
Hindoo numerals, are supposed originally to have been the 
initial letters of the corresponding numeral adjectives. There 
is a marked resemblance between the notation of Ceylon and 
the one used by Aryabhatta in the first chapter of his work, 
and there only. Although the zero and the principle of position 
were unknown to the scholars of Ceylon, they were probably 
known to Aryabhatta; for, in the second chapter, he gives 
directions for extracting the square and cube roots, which 
seem to indicate a knowledge of them. It would appear that 
the zero and the accompanying principle of position were 
introduced about the time of Aryabhatta. These are the 
inventions which give the Hindoo system its great superiority, 
its admirable perfection. 

There appear to have been several notations in use in 
different parts of India, which differed, not in principle, but 
merely in the forms of the signs employed. Of interest is also 


a symbolical system of position, in which the figures generally 
were not expressed by numerical adjectives, but by objects 
suggesting the particular numbers in question. Thus, for 1 
were used the words moon, Brahma, Creator, or form; for 4, 
the words Veda, (because it is divided into four parts) or 
ocean, etc. The following example, taken from the Surya- 
siddhanta, illustrates the idea. The number 1,577,917,828 
is expressed from right to left as follows: Vasu (a class of 
8 gods) + two + eight + mountains (the 7 mountain- chains) 
+ form + digits (the 9 digits) + seven + mountains + lunar 
days (half of which equal 15). The use of such notations 
made it possible to represent a number in several different 
ways. This greatly facilitated the framing of verses containing 
arithmetical rules or scientific constants, which could thus be 
more easily remembered. 

At an early period the Hindoos exhibited great skill in 
calculating, even with large numbers. Thus, they tell us of 
an examination to which Buddha, the reformer of the Indian 
religion, had to submit, when a youth, in order to win the 
maiden he loved. In arithmetic, after having astonished 
his examiners by naming all the periods of numbers up to 
the 53d, he was asked whether he could determine the number 
of primary atoms which, when placed one against the other, 
would form a line one mile in length. Buddha found the 
required answer in this way: 7 primary atoms make a very 
minute grain of dust, 7 of these make a minute grain of dust, 
7 of these a grain of dust whirled up by the wind, and so on. 
Thus he proceeded, step by step, until he finally reached the 


length of a mile. The multiplication of all the factors gave for 
the multitude of primary atoms in a mile a number consisting 
of 15 digits. This problem reminds one of the 'Sand-Counter' 
of Archimedes. 

After the numerical symbolism had been perfected, figuring 
was made much easier. Many of the Indian modes of operation 
differ from ours. The Hindoos were generally inclined to follow 
the motion from left to right, as in writing. Thus, they added 
the left-hand columns first, and made the necessary corrections 
as they proceeded. For instance, they would have added 254 
and 663 thus: 2 + 6 = 8, 5 + 6 = 11, which changes 8 into 9, 
4 + 3 = 7. Hence the sum 917. In subtraction they had two 
methods. Thus in 821 — 348 they would say, 8 from 11 = 3, 

4 from 11 = 7, 3 from 7 = 4. Or they would say, 8 from 11 = 3, 

5 from 12 = 7, 4 from 8 = 4. In multiplication of a number 
by another of only one digit, say 569 by 5, they generally 
said, 5-5 = 25, 5-6 = 30, which changes 25 into 28, 5-9 = 45, 
hence the must be increased by 4. The product is 2845. In 
the multiplication with each other of many-figured numbers, 
they first multiplied, in the manner just indicated, with the 
left-hand digit of the multiplier, which was written above the 
multiplicand, and placed the product above the multiplier. On 
multiplying with the next digit of the multiplier, the product 
was not placed in a new row, as with us, but the first product 
obtained was corrected, as the process continued, by erasing, 
whenever necessary, the old digits, and replacing them by 
new ones, until finally the whole product was obtained. We 
who possess the modern luxuries of pencil and paper, would 



not be likely to fall in love with this Hindoo method. But 
the Indians wrote "with a cane-pen upon a small blackboard 
with a white, thinly liquid paint which made marks that 
could be easily erased, or upon a white tablet, less than a 
foot square, strewn with red flour, on which they wrote the 
figures with a small stick, so that the figures appeared white 
on a red ground." [7] Since the digits had to be quite large 
to be distinctly legible, and since the boards were small, it 
was desirable to have a method which would not require much 
space. Such a one was the above method of multiplication. 
Figures could be easily erased and replaced by others without 
sacrificing neatness. But the Hindoos had also other ways 
of multiplying, of which we mention the following: The 
tablet was divided into squares like 
a chess-board. Diagonals were also 
drawn, as seen in the figure. The 
multiplication of 12 x 735 = 8820 
is exhibited in the adjoining dia- 
gram. [3] The manuscripts extant 8 8 2 
give no information of how divisions 

were executed. The correctness of their additions, sub- 
tractions, and multiplications was tested "by excess of 9's." 
In writing fractions, the numerator was placed above the 
denominator, but no line was drawn between them. 

We shall now proceed to the consideration of some arith- 
metical problems and the Indian modes of solution. A 
favourite method was that of inversion. With laconic brevity, 
Aryabhatta describes it thus: "Multiplication becomes divi- 

/ 7 

/ 3 

/ 5 

/ 4 

/ 6 

/ ° 


sion, division becomes multiplication; what was gain becomes 
loss, what loss, gain; inversion." Quite different from this 
quotation in style is the following problem from Aryabhatta, 
which illustrates the method: [3] "Beautiful maiden with 
beaming eyes, tell me, as thou understandst the right method 
of inversion, which is the number which multiplied by 3, then 
increased by § of the product, divided by 7, diminished by 
j of the quotient, multiplied by itself, diminished by 52, the 
square root extracted, addition of 8, and division by 10, gives 
the number 2?" The process consists in beginning with 2 and 
working backwards. Thus, (2-10 - 8) 2 + 52 = 196, v / 196 = 14, 
and 14-|-7-| 4- 3 = 28, the answer. 

Here is another example taken from Lilavati, a chapter in 
Bhaskara's great work: "The square root of half the number 
of bees in a swarm has flown out upon a jessamine-bush, | of 
the whole swarm has remained behind; one female bee flies 
about a male that is buzzing within a lotus-flower into which 
he was allured in the night by its sweet odour, but is now 
imprisoned in it. Tell me the number of bees." Answer, 72. 
The pleasing poetic garb in which all arithmetical problems 
are clothed is due to the Indian practice of writing all school- 
books in verse, and especially to the fact that these problems, 
propounded as puzzles, were a favourite social amusement. 
Says Brahmagupta: "These problems are proposed simply 
for pleasure; the wise man can invent a thousand others, or 
he can solve the problems of others by the rules given here. 
As the sun eclipses the stars by his brilliancy, so the man of 
knowledge will eclipse the fame of others in assemblies of the 


people if he proposes algebraic problems, and still more if he 
solves them." 

The Hindoos solved problems in interest, discount, part- 
nership, alligation, summation of arithmetical and geometric 
series, devised rules for determining the numbers of combina- 
tions and permutations, and invented magic squares. It may 
here be added that chess, the profoundest of all games, had 
its origin in India. 

The Hindoos made frequent use of the "rule of three," and 
also of the method of "falsa positio," which is almost identical 
with that of the "tentative assumption" of Diophantus. These 
and other rules were applied to a large number of problems. 

Passing now to algebra, we shall first take up the symbols 
of operation. Addition was indicated simply by juxtaposition 
as in Diophantine algebra; subtraction, by placing a dot over 
the subtrahend; multiplication, by putting after the factors, 
bha, the abbreviation of the word bhavita, "the product"; 
division, by placing the divisor beneath the dividend; square- 
root, by writing ka, from the word karana (irrational), 
before the quantity. The unknown quantity was called by 
Brahmagupta ydvattdvat (quantum tanturn). When several 
unknown quantities occurred, he gave, unlike Diophantus, to 
each a distinct name and symbol. The first unknown was 
designated by the general term "unknown quantity." The 
rest were distinguished by names of colours, as the black, 
blue, yellow, red, or green unknown. The initial syllable of 
each word constituted the symbol for the respective unknown 
quantity. Thus yd meant x; ka (from kdlaka = black) meant y; 


yd kd bha, "x times y"; ka 15 ka 10, "yl5 — vlO." 

The Indians were the first to recognise the existence of ab- 
solutely negative quantities. They brought out the difference 
between positive and negative quantities by attaching to the 
one the idea of 'possession,' to the other that of 'debts.' The 
conception also of opposite directions on a line, as an interpre- 
tation of + and — quantities, was not foreign to them. They 
advanced beyond Diophantus in observing that a quadratic 
has always two roots. Thus Bhaskara gives x = 50 and x = — 5 
for the roots of x 1 — 45x = 250. "But," says he, "the second 
value is in this case not to be taken, for it is inadequate; people 
do not approve of negative roots." Commentators speak of 
this as if negative roots were seen, but not admitted. 

Another important generalisation, says Hankel, was this, 
that the Hindoos never confined their arithmetical operations 
to rational numbers. For instance, Bhaskara showed how, by 
the formula 

a + \/a 2 — b a — sja? — b 

r- a T v u, — u 

+ Vb= \ + 

the square root of the sum of rational and irrational numbers 
could be found. The Hindoos never discerned the dividing 
line between numbers and magnitudes, set up by the Greeks, 
which, though the product of a scientific spirit, greatly re- 
tarded the progress of mathematics. They passed from 
magnitudes to numbers and from numbers to magnitudes 
without anticipating that gap which to a sharply discriminat- 
ing mind exists between the continuous and discontinuous. 


Yet by doing so the Indians greatly aided the general progress 
of mathematics. "Indeed, if one understands by algebra the 
application of arithmetical operations to complex magnitudes 
of all sorts, whether rational or irrational numbers or space- 
magnitudes, then the learned Brahmins of Hindostan are the 
real inventors of algebra." [7] 

Let us now examine more closely the Indian algebra. In 
extracting the square and cube roots they used the formulas 
(a + b) 2 = a 2 + 2ab + b 2 and (a + 6) 3 = a 3 + 3a 2 b + 3ab 2 + 6 3 . In 
this connection Aryabhatta speaks of dividing a number into 
periods of two and three digits. From this we infer that the 
principle of position and the zero in the numeral notation were 
already known to him. In figuring with zeros, a statement 
of Bhaskara is interesting. A fraction whose denominator is 
zero, says he, admits of no alteration, though much be added 
or subtracted. Indeed, in the same way, no change takes 
place in the infinite and immutable Deity when worlds are 
destroyed or created, even though numerous orders of beings 
be taken up or brought forth. Though in this he apparently 
evinces clear mathematical notions, yet in other places he 
makes a complete failure in figuring with fractions of zero 

In the Hindoo solutions of determinate equations, Cantor 
thinks he can see traces of Diophantine methods. Some 
technical terms betray their Greek origin. Even if it be true 
that the Indians borrowed from the Greeks, they deserve great 
credit for improving and generalising the solutions of linear 
and quadratic equations. Bhaskara advances far beyond the 


Greeks and even beyond Brahmagupta when he says that "the 
square of a positive, as also of a negative number, is positive; 
that the square root of a positive number is twofold, positive 
and negative. There is no square root of a negative number, 
for it is not a square." Of equations of higher degrees, the 
Indians succeeded in solving only some special cases in which 
both sides of the equation could be made perfect powers by 
the addition of certain terms to each. 

Incomparably greater progress than in the solution of 
determinate equations was made by the Hindoos in the 
treatment of indeterminate equations. Indeterminate analysis 
was a subject to which the Hindoo mind showed a happy 
adaptation. We have seen that this very subject was a 
favourite with Diophantus, and that his ingenuity was almost 
inexhaustible in devising solutions for particular cases. But 
the glory of having invented general methods in this most 
subtle branch of mathematics belongs to the Indians. The 
Hindoo indeterminate analysis differs from the Greek not 
only in method, but also in aim. The object of the former 
was to find all possible integral solutions. Greek analysis, 
on the other hand, demanded not necessarily integral, but 
simply rational answers. Diophantus was content with a 
single solution; the Hindoos endeavoured to find all solutions 
possible. Aryabhatta gives solutions in integers to linear 
equations of the form ax ±by = c, where a, b, c are integers. 
The rule employed is called the pulveriser. For this, as for 
most other rules, the Indians give no proof. Their solution is 
essentially the same as the one of Euler. Euler's process of 


reducing - to a continued fraction amounts to the same as 

the Hindoo process of finding the greatest common divisor of 

a and b by division. This is frequently called the Diophantine 

method. Hankel protests against this name, on the ground 

that Diophantus not only never knew the method, but did 

not even aim at solutions purely integral. [7] These equations 

probably grew out of problems in astronomy. They were 

applied, for instance, to determine the time when a certain 

constellation of the planets would occur in the heavens. 

Passing by the subject of linear equations with more than 
two unknown quantities, we come to indeterminate quadratic 
equations. In the solution of xy = ax + by + c, they applied the 
method re-invented later by Euler, of decomposing (ab + c) 
into the product of two integers m-n and of placing x = m + b 
and y = n + a. 

Remarkable is the Hindoo solution of the quadratic equation 
cy 2 = ax 2 + b. With great keenness of intellect they recognised 
in the special case y 2 = ax 2 + 1 a fundamental problem 
in indeterminate quadratics. They solved it by the cyclic 
method. "It consists," says De Morgan, "in a rule for finding 
an indefinite number of solutions of y 2 = ax 2 + 1 (a being an 
integer which is not a square), by means of one solution given 
or found, and of feeling for one solution by making a solution 
of y 2 = ax 2 + b give a solution of y 2 = ax 2 + b 2 . It amounts to 
the following theorem: If p and q be one set of values of x and y 
in y 2 = ax 2 + b and p' and q' the same or another set, then 
qp + pq' and app' + qq' are values of x and y in y 2 = ax 2 + b 2 . 
From this it is obvious that one solution of y 2 = ax 2 + 1 may 



be made to give any number, and that if, taking b at pleasure, 
y 2 = ax 2 + b 2 can be solved so that x and y are divisible by b, 
then one preliminary solution of y 2 = ax 2 + 1 can be found. 
Another mode of trying for solutions is a combination of the 
preceding with the cuttaca (pulveriser)." These calculations 
were used in astronomy. 

Doubtless this "cyclic method" constitutes the greatest 
invention in the theory of numbers before the time of Lagrange. 
The perversity of fate has willed it, that the equation?/ 2 = ax 2 + 
1 should now be called Pell's problem, while in recognition 
of Brahmin scholarship it ought to be called the "Hindoo 
problem." It is a problem that has exercised the highest 
faculties of some of our greatest modern analysts. By 
them the work of the Hindoos was done over again; for, 
unfortunately, the Arabs transmitted to Europe only a small 
part of Indian algebra and the original Hindoo manuscripts, 
which we now possess, were unknown in the Occident. 

Hindoo geometry is far inferior to the Greek. In it are 
found no definitions, no postulates, no axioms, no logical 
chain of reasoning or rigid form of demonstration, as with 
Euclid. Each theorem stands by itself as an independent 
truth. Like the early Egyptian, it is empirical. Thus, in the 
proof of the theorem of the right triangle, Bhaskara draws the 
right triangle four times 
in the square of the hy- 
potenuse, so that in the 
middle there remains a 
square whose side equals 


the difference between the two sides of the right triangle. 
Arranging this square and the four triangles in a different way, 
they are seen, together, to make up the sum of the square 
of the two sides. "Behold!" says Bhaskara, without adding 
another word of explanation. Bretschneider conjectures that 
the Pythagorean proof was substantially the same as this. 
In another place, Bhaskara gives a second demonstration of 
this theorem by drawing from the vertex of the right angle 
a perpendicular to the hypotenuse, and comparing the two 
triangles thus obtained with the given triangle to which they 
are similar. This proof was unknown in Europe till Wallis re- 
discovered it. The Brahmins never inquired into the properties 
of figures. They considered only metrical relations applicable 
in practical life. In the Greek sense, the Brahmins never had 
a science of geometry. Of interest is the formula given by 
Brahmagupta for the area of a triangle in terms of its sides. 
In the great work attributed to Heron the Elder this formula 
is first found. Whether the Indians themselves invented it, or 
whether they borrowed it from Heron, is a disputed question. 
Several theorems are given by Brahmagupta on quadrilaterals 
which are true only of those which can be inscribed on a 
circle — a limitation which he omits to state. Among these 
is the proposition of Ptolemaeus, that the product of the 
diagonals is equal to the sum of the products of the opposite 
sides. The Hindoos were familiar with the calculation of the 
areas of circles and their segments, of the length of chords 
and perimeters of regular inscribed polygons. An old Indian 
tradition makes n = 3, also = VlO; but Aryabhatta gives the 


value ttwwt- Bhaskara gives two values, — the 'accurate,' ffm, 

1 0Ono ■ i->u.aar^aia gives uwu vtuura, uiic attuiciiic, 12^17 > 

and the 'inaccurate,' Archimedean value, ^. A commentator 
on Lilavati says that these values were calculated by beginning 
with a regular inscribed hexagon, and applying repeatedly 

the formula AD = \j 2 — \J 4 — AB , wherein AB is the side 
of the given polygon, and AD that of one with double the 
number of sides. In this way were obtained the perimeters of 
the inscribed polygons of 12, 24, 48, 96, 192, 384 sides. Taking 
the radius = 100, the perimeter of the last one gives the value 
which Aryabhatta used for -k. 

Greater taste than for geometry was shown by the Hindoos 
for trigonometry. Like the Babylonians and Greeks, they 
divided the circle into quadrants, each quadrant into 90 degrees 
and 5400 minutes. The whole circle was therefore made up 
of 21, 600 equal parts. From Bhaskara's 'accurate' value for n 
it was found that the radius contained 3438 of these circular 
parts. This last step was not Grecian. The Greeks might 
have had scruples about taking a part of a curve as the 
measure of a straight line. Each quadrant was divided into 
24 equal parts, so that each part embraced 225 units of the 
whole circumference, and corresponds to 3| degrees. Notable 
is the fact that the Indians never reckoned, like the Greeks, 
with the whole chord of double the arc, but always with 
the sine (joa) and versed sine. Their mode of calculating 
tables was theoretically very simple. The sine of 90° was 
equal to the radius, or 3438; the sine of 30° was evidently 
half that, or 1719. Applying the formula sin 2 a + cos 2 a = r 2 , 


they obtained sin 45° = \ — = 2431. Substituting for cos a 
its equal sin(90 — a), and making a = 60°, they obtained 

sin 60° = — — = 2978. With the sines of 90, 60, 45, and 30 

as starting-points, they reckoned the sines of half the angles 

by the formula ver sin 2a = 2 sin 2 a, thus obtaining the sines 

of 22° 30', 11° 15', 7° 30', 3° 45'. They now figured out the 

sines of the complements of these angles, namely, the sines of 

86° 15', 82° 30', 78° 45', 75°, 67° 30'; then they calculated the 

sines of half these angles; then of their complements; then, 

again, of half their complements; and so on. By this very 

simple process they got the sines of angles at intervals of 

3° 45'. In this table they discovered the unique law that if 

a, 6, c be three successive arcs such that a — b = b — c = 3° 45', 

sin b 

then sin a — sin b = (sin b — sine) . This formula was 

v ' 225 

afterwards used whenever a re-calculation of tables had to be 

made. No Indian trigonometrical treatise on the triangle is 

extant. In astronomy they solved plane and spherical right 

triangles. [18] 

It is remarkable to what extent Indian mathematics enters 

into the science of our time. Both the form and the spirit 

of the arithmetic and algebra of modern times are essentially 

Indian and not Grecian. Think of that most perfect of 

mathematical symbolisms — the Hindoo notation, think of 

the Indian arithmetical operations nearly as perfect as our 

own, think of their elegant algebraical methods, and then 

judge whether the Brahmins on the banks of the Ganges 

are not entitled to some credit. Unfortunately, some of the 


most brilliant of Hindoo discoveries in indeterminate analysis 
reached Europe too late to exert the influence they would 
have exerted, had they come two or three centuries earlier. 


After the flight of Mohammed from Mecca to Medina in 
622 A.D., an obscure people of Semitic race began to play 
an important part in the drama of history. Before the lapse 
of ten years, the scattered tribes of the Arabian peninsula 
were fused by the furnace blast of religious enthusiasm into 
a powerful nation. With sword in hand the united Arabs 
subdued Syria and Mesopotamia. Distant Persia and the 
lands beyond, even unto India, were added to the dominions 
of the Saracens. They conquered Northern Africa, and nearly 
the whole Spanish peninsula, but were finally checked from 
further progress in Western Europe by the firm hand of 
Charles Martel (732 A.D.). The Moslem dominion extended 
now from India to Spain; but a war of succession to the 
caliphate ensued, and in 755 the Mohammedan empire was 
divided, — one caliph reigning at Bagdad, the other at Cordova 
in Spain. Astounding as was the grand march of conquest by 
the Arabs, still more so was the ease with which they put aside 
their former nomadic life, adopted a higher civilisation, and 
assumed the sovereignty over cultivated peoples. Arabic was 
made the written language throughout the conquered lands. 
With the rule of the Abbasides in the East began a new period 
in the history of learning. The capital, Bagdad, situated 


on the Euphrates, lay half-way between two old centres of 
scientific thought, — India in the East, and Greece in the West. 
The Arabs were destined to be the custodians of the torch 
of Greek and Indian science, to keep it ablaze during the 
period of confusion and chaos in the Occident, and afterwards 
to pass it over to the Europeans. Thus science passed from 
Aryan to Semitic races, and then back again to the Aryan. 
The Mohammedans have added but little to the knowledge 
in mathematics which they received. They now and then 
explored a small region to which the path had been previously 
pointed out, but they were quite incapable of discovering new 
fields. Even the more elevated regions in which the Hellenes 
and Hindoos delighted to wander — namely, the Greek conic 
sections and the Indian indeterminate analysis — were seldom 
entered upon by the Arabs. They were less of a speculative, 
and more of a practical turn of mind. 

The Abbasides at Bagdad encouraged the introduction 
of the sciences by inviting able specialists to their court, 
irrespective of nationality or religious belief. Medicine and 
astronomy were their favourite sciences. Thus Haroun-al- 
Raschid, the most distinguished Saracen ruler, drew Indian 
physicians to Bagdad. In the year 772 there came to the court 
of Caliph Almansur a Hindoo astronomer with astronomical 
tables which were ordered to be translated into Arabic. These 
tables, known by the Arabs as the Sindhind, and probably 
taken from the Brahma-sphuta-siddhanta of Brahmagupta, 
stood in great authority. They contained the important 
Hindoo table of sines. 


Doubtless at this time, and along with these astronomical 
tables, the Hindoo numerals, with the zero and the principle 
of position, were introduced among the Saracens. Before the 
time of Mohammed the Arabs had no numerals. Numbers 
were written out in words. Later, the numerous computa- 
tions connected with the financial administration over the 
conquered lands made a short symbolism indispensable. In 
some localities, the numerals of the more civilised conquered 
nations were used for a time. Thus in Syria, the Greek 
notation was retained; in Egypt, the Coptic. In some cases, 
the numeral adjectives may have been abbreviated in writing. 
The Diwani-numerals, found in an Arabic- Persian dictionary, 
are supposed to be such abbreviations. Gradually it became 
the practice to employ the 28 Arabic letters of the alphabet for 
numerals, in analogy to the Greek system. This notation was 
in turn superseded by the Hindoo notation, which quite early 
was adopted by merchants, and also by writers on arithmetic. 
Its superiority was so universally recognised, that it had no 
rival, except in astronomy, where the alphabetic notation 
continued to be used. Here the alphabetic notation offered 
no great disadvantage, since in the sexagesimal arithmetic, 
taken from the Almagest, numbers of generally only one or 
two places had to be written. [7] 

As regards the form of the so-called Arabic numerals, the 
statement of the Arabic writer Albiruni (died 1039), who 
spent many years in India, is of interest. He says that the 
shape of the numerals, as also of the letters in India, differed 
in different localities, and that the Arabs selected from the 


various forms the most suitable. An Arabian astronomer 
says there was among people much difference in the use 
of symbols, especially of those for 5, 6, 7, and 8. The 
symbols used by the Arabs can be traced back to the tenth 
century. We find material differences between those used by 
the Saracens in the East and those used in the West. But most 
surprising is the fact that the symbols of both the East and 
of the West Arabs deviate so extraordinarily from the Hindoo 
Devanagari numerals (= divine numerals) of to-day, and that 
they resemble much more closely the apices of the Roman 
writer Boethius. This strange similarity on the one hand, and 
dissimilarity on the other, is difficult to explain. The most 
plausible theory is the one of Woepcke: (1) that about the 
second century after Christ, before the zero had been invented, 
the Indian numerals were brought to Alexandria, whence they 
spread to Rome and also to West Africa; (2) that in the eighth 
century, after the notation in India had been already much 
modified and perfected by the invention of the zero, the Arabs 
at Bagdad got it from the Hindoos; (3) that the Arabs of 
the West borrowed the Columbus-egg, the zero, from those 
in the East, but retained the old forms of the nine numerals, 
if for no other reason, simply to be contrary to their political 
enemies of the East; (4) that the old forms were remembered 
by the West- Arabs to be of Indian origin, and were hence 
called Gubar-numerals (= dust-numerals, in memory of the 
Brahmin practice of reckoning on tablets strewn with dust or 
sand; (5) that, since the eighth century, the numerals in India 
underwent further changes, and assumed the greatly modified 


forms of the modern Devanagari- numerals. [3] This is rather 
a bold theory, but, whether true or not, it explains better than 
any other yet propounded, the relations between the apices, 
the Gubar, the East- Arabic, and Devanagari numerals. 

It has been mentioned that in 772 the Indian Siddhanta was 
brought to Bagdad and there translated into Arabic. There is 
no evidence that any intercourse existed between Arabic and 
Indian astronomers either before or after this time, excepting 
the travels of Albiruni. But we should be very slow to deny 
the probability that more extended communications actually 
did take place. 

Better informed are we regarding the way in which Greek 
science, in successive waves, dashed upon and penetrated 
Arabic soil. In Syria the sciences, especially philosophy and 
medicine, were cultivated by Greek Christians. Celebrated 
were the schools at Antioch and Emesa, and, first of all, the 
flourishing Nestorian school at Edessa. From Syria, Greek 
physicians and scholars were called to Bagdad. Translations 
of works from the Greek began to be made. A large number 
of Greek manuscripts were secured by Caliph Al Mamun 
(813-833) from the emperor in Constantinople and were 
turned over to Syria. The successors of Al Mamun continued 
the work so auspiciously begun, until, at the beginning of 
the tenth century, the more important philosophic, medical, 
mathematical, and astronomical works of the Greeks could 
all be read in the Arabic tongue. The translations of 
mathematical works must have been very deficient at first, 
as it was evidently difficult to secure translators who were 


masters of both the Greek and Arabic and at the same time 
proficient in mathematics. The translations had to be revised 
again and again before they were satisfactory. The first Greek 
authors made to speak in Arabic were Euclid and Ptolemaeus. 
This was accomplished during the reign of the famous Haroun- 
al-Raschid. A revised translation of Euclid's Elements was 
ordered by Al Mamun. As this revision still contained 
numerous errors, a new translation was made, either by the 
learned Honein ben Ishak, or by his son, Ishak ben Honein. To 
the thirteen books of the Elements were added the fourteenth, 
written by Hypsicles, and the fifteenth by Damascius. But it 
remained for Tabit ben Korra to bring forth an Arabic Euclid 
satisfying every need. Still greater difficulty was experienced 
in securing an intelligible translation of the Almagest. Among 
other important translations into Arabic were the works of 
Apollonius, Archimedes, Heron, and Diophantus. Thus we 
see that in the course of one century the Arabs gained access 
to the vast treasures of Greek science. Having been little 
accustomed to abstract thought, we need not marvel if, during 
the ninth century, all their energy was exhausted merely in 
appropriating the foreign material. No attempts were made 
at original work in mathematics until the next century. 

In astronomy, on the other hand, great activity in original 
research existed as early as the ninth century. The religious 
observances demanded by Mohammedanism presented to as- 
tronomers several practical problems. The Moslem dominions 
being of such enormous extent, it remained in some localities 
for the astronomer to determine which way the "Believer" 


must turn during prayer that he may be facing Mecca. The 
prayers and ablutions had to take place at definite hours during 
the day and night. This led to more accurate determinations 
of time. To fix the exact date for the Mohammedan feasts it 
became necessary to observe more closely the motions of the 
moon. In addition to all this, the old Oriental superstition 
that extraordinary occurrences in the heavens in some myste- 
rious way affect the progress of human affairs added increased 
interest to the prediction of eclipses. [7] 

For these reasons considerable progress was made. Astro- 
nomical tables and instruments were perfected, observatories 
erected, and a connected series of observations instituted. 
This intense love for astronomy and astrology continued 
during the whole Arabic scientific period. As in India, so 
here, we hardly ever find a man exclusively devoted to pure 
mathematics. Most of the so-called mathematicians were first 
of all astronomers. 

The first notable author of mathematical books was Mo- 
hammed ben Musa Al Hovarezmi, who lived during the 
reign of Caliph Al Mamun (813-833). He was engaged by the 
caliph in making extracts from the Sindhind, in revising the 
tablets of Ptolemaeus, in taking observations at Bagdad and 
Damascus, and in measuring a degree of the earth's meridian. 
Important to us is his work on algebra and arithmetic. The 
portion on arithmetic is not extant in the original, and it was 
not till 1857 that a Latin translation of it was found. It begins 
thus: "Spoken has Algoritmi. Let us give deserved praise to 
God, our leader and defender." Here the name of the author, 


Al Hovarezmi, has passed into Algoritmi, from which comes 

our modern word, algorithm, signifying the art of computing 

in any particular way. The arithmetic of Hovarezmi, being 

based on the principle of position and the Hindoo method 

of calculation, "excels," says an Arabic writer, "all others 

in brevity and easiness, and exhibits the Hindoo intellect 

and sagacity in the grandest inventions." This book was 

followed by a large number of arithmetics by later authors, 

which differed from the earlier ones chiefly in the greater 

variety of methods. Arabian arithmetics generally contained 

the four operations with integers and fractions, modelled 

after the Indian processes. They explained the operation of 

casting out the 9's, which was sometimes called the "Hindoo 

proof." They contained also the regula falsa and the regula 

duorum falsorum, by which algebraical examples could be 

solved without algebra. Both these methods were known to 

the Indians. The regula falsa or falsa positio was the assigning 

of an assumed value to the unknown quantity, which value, 

if wrong, was corrected by some process like the "rule of 

three." Diophantus used a method almost identical with this. 

The regula duorum falsorum was as follows: [7] To solve an 

equation f(x) = V, assume, for the moment, two values for x; 

namely, x = a and x = b. Then form f(a) = A and f(b) = B, 

and determine the errors V — A = E a and V — B = E ; then the 

required x = — is generally a close approximation, 

E a — Ei, 

but is absolutely accurate whenever f(x) is a linear function 
of x. 

We now return to Hovarezmi, and consider the other part 


of his work, — the algebra. This is the first book known to 
contain this word itself as title. Really the title consists of two 
words, aldshebr walmukabala, the nearest English translation 
of which is "restoration" and "reduction." By "restoration" 
was meant the transposing of negative terms to the other side 
of the equation; by "reduction," the uniting of similar terms. 
Thus, x 2 — 2x = 5x + 6 passes by aldshebr into x 1 = 5x + 2x + 6; 
and this, by walmukabala, into x 1 = 7x + 6. The work on 
algebra, like the arithmetic, by the same author, contains 
nothing original. It explains the elementary operations and 
the solutions of linear and quadratic equations. From whom 
did the author borrow his knowledge of algebra? That it came 
entirely from Indian sources is impossible, for the Hindoos had 
no such rules like the "restoration" and "reduction." They 
were, for instance, never in the habit of making all terms in an 
equation positive, as is done by the process of "restoration." 
Diophantus gives two rules which resemble somewhat those of 
our Arabic author, but the probability that the Arab got all 
his algebra from Diophantus is lessened by the considerations 
that he recognised both roots of a quadratic, while Diophantus 
noticed only one; and that the Greek algebraist, unlike the 
Arab, habitually rejected irrational solutions. It would seem, 
therefore, that the algebra of Hovarezmi was neither purely 
Indian nor purely Greek, but was a hybrid of the two, with 
the Greek element predominating. 

The algebra of Hovarezmi contains also a few meagre 
fragments on geometry. He gives the theorem of the right 
triangle, but proves it after Hindoo fashion and only for the 


simplest case, when the right triangle is isosceles. He then 
calculates the areas of the triangle, parallelogram, and circle. 
For 7r he uses the value 3y, and also the two Indian, n = a/10 
and 7r = |qqqq . Strange to say, the last value was afterwards 
forgotten by the Arabs, and replaced by others less accurate. 
This bit of geometry doubtless came from India. Later Arabic 
writers got their geometry almost entirely from Greece. 

Next to be noticed are the three sons of Musa ben 
Sakir, who lived in Bagdad at the court of the Caliph Al 
Mamun. They wrote several works, of which we mention a 
geometry in which is also contained the well-known formula 
for the area of a triangle expressed in terms of its sides. We 
are told that one of the sons travelled to Greece, probably 
to collect astronomical and mathematical manuscripts, and 
that on his way back he made acquaintance with Tabit ben 
Korra. Recognising in him a talented and learned astronomer, 
Mohammed procured for him a place among the astronomers 
at the court in Bagdad. Tabit ben Korra (836-901) was 
born at Harran in Mesopotamia. He was proficient not 
only in astronomy and mathematics, but also in the Greek, 
Arabic, and Syrian languages. His translations of Apollonius, 
Archimedes, Euclid, Ptolemy, Theodosius, rank among the 
best. His dissertation on amicable numbers (of which each is 
the sum of the factors of the other) is the first known specimen 
of original work in mathematics on Arabic soil. It shows 
that he was familiar with the Pythagorean theory of numbers. 
Tabit invented the following rule for finding amicable numbers: 
lip = 3-2 n -l,g = 3-2 n_1 - 1, r = 9-2 2 ™- 1 - 1 (n being a whole 


number) are three primes, then a = 2 n pq, b = 2 n r are a pair of 
amicable numbers. Thus, if n = 2, then p = 11, q = 5, r = 71, 
and a = 220, b = 284. Tabit also trisected an angle. 

Foremost among the astronomers of the ninth century 
ranked Al Battani, called Albategnius by the Latins. Battan 
in Syria was his birthplace. His observations were celebrated 
for great precision. His work, De scientia stellarum, was 
translated into Latin by Plato Tiburtinus, in the twelfth 
century. Out of this translation sprang the word 'sinus,' as the 
name of a trigonometric function. The Arabic word for "sine," 
dschiba, was derived from the Sanscrit jiva, and resembled 
the Arabic word dschaib, meaning an indentation or gulf. 
Hence the Latin "sinus." [3] Al Battani was a close student 
of Ptolemy, but did not follow him altogether. He took an 
important step for the better, when he introduced the Indian 
"sine" or half the chord, in place of the whole chord of Ptolemy. 
Another improvement on Greek trigonometry made by the 
Arabs points likewise to Indian influences. Propositions and 
operations which were treated by the Greeks geometrically 

are expressed by the Arabs algebraically. Thus, Al Battani 

r ■ sin 9 

at once gets from an equation = D, the value ol 6 

jj cos 6 

by means of sin# = , — a process unknown to the 

VI + D 2 
ancients. He knows, of course, all the formulas for spherical 

triangles given in the Almagest, but goes further, and adds an 

important one of his own for oblique-angled triangles; namely, 

cos a = cos b cos c + sin b sin c cos A. 

At the beginning of the tenth century political troubles 


arose in the East, and as a result the house of the Abbasides 
lost power. One province after another was taken, till, in 945, 
all possessions were wrested from them. Fortunately, the 
new rulers at Bagdad, the Persian Buyides, were as much 
interested in astronomy as their predecessors. The progress 
of the sciences was not only unchecked, but the conditions 
for it became even more favourable. The Emir Adud-ed-daula 
(978-983) gloried in having studied astronomy himself. His 
son Saraf-ed-daula erected an observatory in the garden of 
his palace, and called thither a whole group of scholars. [7] 
Among them were Abul Wefa, Al Kuhi, Al Sagani. 

Abul Wefa (940-998) was born at Buzshan in Chorassan, 
a region among the Persian mountains, which has brought 
forth many Arabic astronomers. He forms an important 
exception to the unprogressive spirit of Arabian scientists 
by his brilliant discovery of the variation of the moon, an 
inequality usually supposed to have been first discovered by 
Tycho Brahe. [11] Abul Wefa translated Diophantus. He 
is one of the last Arabic translators and commentators of 
Greek authors. The fact that he esteemed the algebra of 
Mohammed ben Musa Hovarezmi worthy of his commentary 
indicates that thus far algebra had made little or no progress 
on Arabic soil. Abul Wefa invented a method for computing 
tables of sines which gives the sine of half a degree correct 
to nine decimal places. He did himself credit by introducing 
the tangent into trigonometry and by calculating a table of 
tangents. The first step toward this had been taken by Al 
Battani. Unfortunately, this innovation and the discovery of 


the moon's variation excited apparently no notice among his 
contemporaries and followers. "We can hardly help looking 
upon this circumstance as an evidence of a servility of intellect 
belonging to the Arabian period." A treatise by Abul Wefa 
on "geometric constructions" indicates that efforts were being 
made at that time to improve draughting. It contains a neat 
construction of the corners of the regular polyhedrons on 
the circumscribed sphere. Here, for the first time, appears 
the condition which afterwards became very famous in the 
Occident, that the construction be effected with a single 
opening of the compass. 

Al Kuhi, the second astronomer at the observatory of 
the emir at Bagdad, was a close student of Archimedes and 
Apollonius. He solved the problem, to construct a segment of a 
sphere equal in volume to a given segment and having a curved 
surface equal in area to that of another given segment. He, 
Al Sagani, and Al Biruni made a study of the trisection of 
angles. Abul Gud, an able geometer, solved the problem by 
the intersection of a parabola with an equilateral hyperbola. 

The Arabs had already discovered the theorem that the 
sum of two cubes can never be a cube. Abu Mohammed 
Al Hogendi of Chorassan thought he had proved this, but 
we are told that the demonstration was defective. Creditable 
work in theory of numbers and algebra was done by Al Karhi 
of Bagdad, who lived at the beginning of the eleventh century. 
His treatise on algebra is the greatest algebraic work of the 
Arabs. In it he appears as a disciple of Diophantus. He was 
the first to operate with higher roots and to solve equations 


of the form x 2n + ax n = b. For the solution of quadratic 
equations he gives both arithmetical and geometric proofs. 
He was the first Arabic author to give and prove the theorems 
on the summation of the series:— 

l 2 + 2 2 + 3 2 + --- + n 2 = (i + 2 + ... + n) " + 


l 3 + 2 3 + 3 3 + • • • + n 3 = (1 + 2 + • • • + n) 2 . 

Al Karhi also busied himself with indeterminate analysis. 
He showed skill in handling the methods of Diophantus, but 
added nothing whatever to the stock of knowledge already 
on hand. As a subject for original research, indeterminate 
analysis was too subtle for even the most gifted of Arabian 
minds. Rather surprising is the fact that Al Karhi's algebra 
shows no traces whatever of Hindoo indeterminate analysis. 
But most astonishing it is, that an arithmetic by the same 
author completely excludes the Hindoo numerals. It is 
constructed wholly after Greek pattern. Abul Wefa also, in 
the second half of the tenth century, wrote an arithmetic in 
which Hindoo numerals find no place. This practice is the 
very opposite to that of other Arabian authors. The question, 
why the Hindoo numerals were ignored by so eminent authors, 
is certainly a puzzle. Cantor suggests that at one time there 
may have been rival schools, of which one followed almost 
exclusively Greek mathematics, the other Indian. 

The Arabs were familiar with geometric solutions of quad- 
ratic equations. Attempts were now made to solve cubic 
equations geometrically. They were led to such solutions 


by the study of questions like the Archimedean problem, 
demanding the section of a sphere by a plane so that the two 
segments shall be in a prescribed ratio. The first to state 
this problem in form of a cubic equation was Al Mahani of 
Bagdad, while Abu Gafar Al Hazin was the first Arab to 
solve the equation by conic sections. Solutions were given 
also by Al Kuhi, Al Hasan ben Al Haitam, and others. [20] 
Another difficult problem, to determine the side of a regular 
heptagon, required the construction of the side from the 
equation x 3 - x 2 - 2x + 1 = 0. It was attempted by many and 
at last solved by Abul Gud. 

The one who did most to elevate to a method the solution 
of algebraic equations by intersecting conies, was Omar al 
Hayyami of Chorassan, about 1079 A.D. He divides cubics 
into two classes, the trinomial and quadrinomial, and each class 
into families and species. Each species is treated separately 
but according to a general plan. He believed that cubics could 
not be solved by calculation, nor biquadratics by geometry. 
He rejected negative roots and often failed to discover all 
the positive ones. Attempts at biquadratic equations were 
made by Abul Wefa, [20] who solved geometrically x 4 = a and 
x + aar = b. 

The solution of cubic equations by intersecting conies was 
the greatest achievement of the Arabs in algebra. The 
foundation to this work had been laid by the Greeks, for it 
was Menaechmus who first constructed the roots of x 3 — a = 
or x 3 — 2a 3 = 0. It was not his aim to find the number 
corresponding to x, but simply to determine the side x of 


a cube double another cube of side a. The Arabs, on the 
other hand, had another object in view: to find the roots 
of given numerical equations. In the Occident, the Arabic 
solutions of cubics remained unknown until quite recently. 
Descartes and Thomas Baker invented these constructions 
anew. The works of Al Hayyami, Al Karhi, Abul Gud, show 
how the Arabs departed further and further from the Indian 
methods, and placed themselves more immediately under 
Greek influences. In this way they barred the road of progress 
against themselves. The Greeks had advanced to a point 
where material progress became difficult with their methods; 
but the Hindoos furnished new ideas, many of which the Arabs 
now rejected. 

With Al Karhi and Omar Al Hayyami, mathematics among 
the Arabs of the East reached flood-mark, and now it begins 
to ebb. Between 1100 and 1300 A.D. come the crusades 
with war and bloodshed, during which European Christians 
profited much by their contact with Arabian culture, then 
far superior to their own; but the Arabs got no science from 
the Christians in return. The crusaders were not the only 
adversaries of the Arabs. During the first half of the thirteenth 
century, they had to encounter the wild Mongolian hordes, 
and, in 1256, were conquered by them under the leadership of 
Hulagu. The caliphate at Bagdad now ceased to exist. At the 
close of the fourteenth century still another empire was formed 
by Timur or Tamerlane, the Tartar. During such sweeping 
turmoil, it is not surprising that science declined. Indeed, 
it is a marvel that it existed at all. During the supremacy 


of Hulagu, lived Nasir Eddin (1201-1274), a man of broad 
culture and an able astronomer. He persuaded Hulagu to 
build him and his associates a large observatory at Maraga. 
Treatises on algebra, geometry, arithmetic, and a translation 
of Euclid's Elements, were prepared by him. Even at the 
court of Tamerlane in Samarkand, the sciences were by no 
means neglected. A group of astronomers was drawn to this 
court. Ulug Beg (1393-1449), a grandson of Tamerlane, was 
himself an astronomer. Most prominent at this time was Al 
Kaschi, the author of an arithmetic. Thus, during intervals 
of peace, science continued to be cultivated in the East for 
several centuries. The last Oriental writer was Beha Eddin 
(1547-1622). His Essence of Arithmetic stands on about the 
same level as the work of Mohammed ben Musa Hovarezmi, 
written nearly 800 years before. 

"Wonderful is the expansive power of Oriental peoples, with 
which upon the wings of the wind they conquer half the world, 
but more wonderful the energy with which, in less than two 
generations, they raise themselves from the lowest stages of 
cultivation to scientific efforts." During all these centuries, 
astronomy and mathematics in the Orient greatly excel these 
sciences in the Occident. 

Thus far we have spoken only of the Arabs in the East. 
Between the Arabs of the East and of the West, which were 
under separate governments, there generally existed consid- 
erable political animosity. In consequence of this, and of the 
enormous distance between the two great centres of learning, 
Bagdad and Cordova, there was less scientific intercourse 


among them than might be expected to exist between peoples 
having the same religion and written language. Thus the 
course of science in Spain was quite independent of that 
in Persia. While wending our way westward to Cordova, 
we must stop in Egypt long enough to observe that there, 
too, scientific activity was rekindled. Not Alexandria, but 
Cairo with its library and observatory, was now the home 
of learning. Foremost among her scientists ranked Ben 
Junus (died 1008), a contemporary of Abul Wefa. He solved 
some difficult problems in spherical trigonometry. Another 
Egyptian astronomer was Ibn Al Haitam (died 1038), who 
wrote on geometric loci. Travelling westward, we meet in 
Morocco Abul Hasan Ali, whose treatise 'on astronomical 
instruments' discloses a thorough knowledge of the Conies of 
Apollonius. Arriving finally in Spain at the capital, Cordova, 
we are struck by the magnificent splendour of her architecture. 
At this renowned seat of learning, schools and libraries were 
founded during the tenth century. 

Little is known of the progress of mathematics in Spain. The 
earliest name that has come down to us is Al Madshriti (died 
1007), the author of a mystic paper on 'amicable numbers.' 
His pupils founded schools at Cordova, Dania, and Granada. 
But the only great astronomer among the Saracens in Spain is 
Gabir ben Aflah of Sevilla, frequently called Geber. He lived 
in the second half of the eleventh century. It was formerly 
believed that he was the inventor of algebra, and that the 
word algebra came from 'Gabir' or 'Geber.' He ranks among 
the most eminent astronomers of this time, but, like so many 


of his contemporaries, his writings contain a great deal of 
mysticism. His chief work is an astronomy in nine books, of 
which the first is devoted to trigonometry. In his treatment 
of spherical trigonometry, he exercises great independence of 
thought. He makes war against the time-honoured procedure 
adopted by Ptolemy of applying "the rule of six quantities," 
and gives a new way of his own, based on the 'rule of four 
quantities.' This is: If PP\ and QQi be two arcs of great circles 
intersecting in A, and if PQ and P\Qi be arcs of great circles 
drawn perpendicular to QQ\, then we have the proportion 

sin^4P : sin PQ = sin^P^ : sinPiQi. 

From this he derives the formulas for spherical right triangles. 
To the four fundamental formulas already given by Ptolemy, 
he added a fifth, discovered by himself. If a, b, c, be the 
sides, and A, B, C, the angles of a spherical triangle, right- 
angled at A, then cosB = cosbs'mC. This is frequently called 
"Geber's Theorem." Radical and bold as were his innovations 
in spherical trigonometry, in plane trigonometry he followed 
slavishly the old beaten path of the Greeks. Not even did he 
adopt the Indian 'sine' and 'cosine,' but still used the Greek 
'chord of double the angle.' So painful was the departure 
from old ideas, even to an independent Arab! After the 
time of Gabir ben Aflah there was no mathematician among 
the Spanish Saracens of any reputation. In the year in 
which Columbus discovered America, the Moors lost their last 
foothold on Spanish soil. 

We have witnessed a laudable intellectual activity among 


the Arabs. They had the good fortune to possess rulers who, 
by their munificence, furthered scientific research. At the 
courts of the caliphs, scientists were supplied with libraries 
and observatories. A large number of astronomical and 
mathematical works were written by Arabic authors. Yet 
we fail to find a single important principle in mathematics 
brought forth by the Arabic mind. Whatever discoveries 
they made, were in fields previously traversed by the Greeks 
or the Indians, and consisted of objects which the latter 
had overlooked in their rapid march. The Arabic mind 
did not possess that penetrative insight and invention by 
which mathematicians in Europe afterwards revolutionised 
the science. The Arabs were learned, but not original. Their 
chief service to science consists in this, that they adopted the 
learning of Greece and India, and kept what they received 
with scrupulous care. When the love for science began to 
grow in the Occident, they transmitted to the Europeans the 
valuable treasures of antiquity. Thus a Semitic race was, 
during the Dark Ages, the custodian of the Aryan intellectual 


With the third century after Christ begins an era of 
migration of nations in Europe. The powerful Goths quit their 
swamps and forests in the North and sweep onward in steady 
southwestern current, dislodging the Vandals, Sueves, and 
Burgundians, crossing the Roman territory, and stopping and 


recoiling only when reaching the shores of the Mediterranean. 
From the Ural Mountains wild hordes sweep down on the 
Danube. The Roman Empire falls to pieces, and the Dark 
Ages begin. But dark though they seem, they are the 
germinating season of the institutions and nations of modern 
Europe. The Teutonic element, partly pure, partly intermixed 
with the Celtic and Latin, produces that strong and luxuriant 
growth, the modern civilisation of Europe. Almost all the 
various nations of Europe belong to the Aryan stock. As the 
Greeks and the Hindoos — both Aryan races — were the great 
thinkers of antiquity, so the nations north of the Alps became 
the great intellectual leaders of modern times. 

Introduction of Roman Mathematics. 

We shall now consider how these as yet barbaric nations 
of the North gradually came in possession of the intellectual 
treasures of antiquity. With the spread of Christianity 
the Latin language was introduced not only in ecclesiastical 
but also in scientific and all important worldly transactions. 
Naturally the science of the Middle Ages was drawn largely 
from Latin sources. In fact, during the earlier of these ages 
Roman authors were the only ones read in the Occident. 
Though Greek was not wholly unknown, yet before the 
thirteenth century not a single Greek scientific work had been 
read or translated into Latin. Meagre indeed was the science 
which could be gotten from Roman writers, and we must wait 
several centuries before any substantial progress is made in 



After the time of Boethius and Cassiodorius mathematical 
activity in Italy died out. The first slender blossom of 
science among tribes that came from the North was an 
encyclopaedia entitled Origines, written by Isidorus (died 
636 as bishop of Seville). This work is modelled after the 
Roman encyclopaedias of Martianus Capella of Carthage and 
of Cassiodorius. Part of it is devoted to the quadrivium, 
arithmetic, music, geometry, and astronomy. He gives 
definitions and grammatical explications of technical terms, 
but does not describe the modes of computation then in vogue. 
After Isidorus there follows a century of darkness which is at 
last dissipated by the appearance of Bede the Venerable 
(672-735) , the most learned man of his time. He was a native of 
Ireland, then the home of learning in the Occident. His works 
contain treatises on the Computus, or the computation of 
Easter-time, and on finger-reckoning. It appears that a finger- 
symbolism was then widely used for calculation. The correct 
determination of the time of Easter was a problem which in 
those days greatly agitated the Church. It became desirable 
to have at least one monk at each monastery who could 
determine the day of religious festivals and could compute the 
calendar. Such determinations required some knowledge of 
arithmetic. Hence we find that the art of calculating always 
found some little corner in the curriculum for the education 
of monks. 

The year in which Bede died is also the year in which Alcuin 
(735-804) was born. Alcuin was educated in Ireland, and was 


called to the court of Charlemagne to direct the progress of 
education in the great Frankish Empire. Charlemagne was 
a great patron of learning and of learned men. In the great 
sees and monasteries he founded schools in which were taught 
the psalms, writing, singing, computation (computus), and 
grammar. By computus was here meant, probably, not merely 
the determination of Easter-time, but the art of computation 
in general. Exactly what modes of reckoning were then 
employed we have no means of knowing. It is not likely that 
Alcuin was familiar with the apices of Boethius or with the 
Roman method of reckoning on the abacus. He belongs to 
that long list of scholars who dragged the theory of numbers 
into theology. Thus the number of beings created by God, 
who created all things well, is 6, because 6 is a perfect number 
(the sum of its divisors being 1 + 2 + 3 = 6); 8, on the other 
hand, is an imperfect number (1 + 2 + 4 < 8); hence the second 
origin of mankind emanated from the number 8, which is the 
number of souls said to have been in Noah's ark. 

There is a collection of "Problems for Quickening the Mind" 
(propositiones ad acuendos iuvenes), which are certainly as old 
as 1000 A.D. and possibly older. Cantor is of the opinion that 
they were written much earlier and by Alcuin. The following 
is a specimen of these "Problems": A dog chasing a rabbit, 
which has a start of 150 feet, jumps 9 feet every time the rabbit 
jumps 7. In order to determine in how many leaps the dog 
overtakes the rabbit, 150 is to be divided by 2. In this collection 
of problems, the areas of triangular and quadrangular pieces 
of land are found by the same formulas of approximation as 


those used by the Egyptians and given by Boethius in his 
geometry. An old problem is the "cistern-problem" (given 
the time in which several pipes can fill a cistern singly, to find 
the time in which they fill it jointly), which has been found 
previously in Heron, in the Greek Anthology, and in Hindoo 
works. Many of the problems show that the collection was 
compiled chiefly from Roman sources. The problem which, on 
account of its uniqueness, gives the most positive testimony 
regarding the Roman origin is that on the interpretation of a 
will in a case where twins are born. The problem is identical 
with the Roman, except that different ratios are chosen. Of 
the exercises for recreation, we mention the one of the wolf, 
goat, and cabbage, to be rowed across a river in a boat 
holding only one besides the ferry-man. Query: How must he 
carry them across so that the goat shall not eat the cabbage, 
nor the wolf the goat? The solutions of the "problems for 
quickening the mind" require no further knowledge than the 
recollection of some few formulas used in surveying, the ability 
to solve linear equations and to perform the four fundamental 
operations with integers. Extraction of roots was nowhere 
demanded; fractions hardly ever occur. [3] 

The great empire of Charlemagne tottered and fell almost 
immediately after his death. War and confusion ensued. 
Scientific pursuits were abandoned, not to be resumed until 
the close of the tenth century, when under Saxon rule in 
Germany and Capetian in France, more peaceful times began. 
The thick gloom of ignorance commenced to disappear. The 
zeal with which the study of mathematics was now taken up 


by the monks is due principally to the energy and influence of 
one man, — Gerbert. He was born in Aurillac in Auvergne. 
After receiving a monastic education, he engaged in study, 
chiefly of mathematics, in Spain. On his return he taught 
school at Rheims for ten years and became distinguished for 
his profound scholarship. By King Otto I. and his successors 
Gerbert was held in highest esteem. He was elected bishop 
of Rheims, then of Ravenna, and finally was made Pope 
under the name of Sylvester II. by his former pupil Emperor 
Otho III. He died in 1003, after a life intricately involved 
in many political and ecclesiastical quarrels. Such was the 
career of the greatest mathematician of the tenth century in 
Europe. By his contemporaries his mathematical knowledge 
was considered wonderful. Many even accused him of criminal 
intercourse with evil spirits. 

Gerbert enlarged the stock of his knowledge by procuring 
copies of rare books. Thus in Mantua he found the geometry 
of Boethius. Though this is of small scientific value, yet 
it is of great importance in history. It was at that time 
the only book from which European scholars could learn the 
elements of geometry. Gerbert studied it with zeal, and is 
generally believed himself to be the author of a geometry. 
H. Weissenborn denies his authorship, and claims that the 
book in question consists of three parts which cannot come 
from one and the same author. [21] This geometry contains 
nothing more than the one of Boethius, but the fact that 
occasional errors in the latter are herein corrected shows that 
the author had mastered the subject. "The first mathematical 


paper of the Middle Ages which deserves this name," says 
Hankel, "is a letter of Gerbert to Adalbold, bishop of 
Utrecht," in which is explained the reason why the area of a 
triangle, obtained "geometrically" by taking the product of 
the base by half its altitude, differs from the area calculated 
"arithmetically," according to the formula \a(a + 1), used by 
surveyors, where a stands for a side of an equilateral triangle. 
He gives the correct explanation that in the latter formula 
all the small squares, in which the triangle is supposed to be 
divided, are counted in wholly, even though parts of them 
project beyond it. 

Gerbert made a careful study of the arithmetical works 
of Boethius. He himself published two works, — Rule of 
Computation on the Abacus, and A Small Book on the 
Division of Numbers. They give an insight into the methods 
of calculation practised in Europe before the introduction 
of the Hindoo numerals. Gerbert used the abacus, which 
was probably unknown to Alcuin. Bernelinus, a pupil of 
Gerbert, describes it as consisting of a smooth board upon 
which geometricians were accustomed to strew blue sand, and 
then to draw their diagrams. For arithmetical purposes the 
board was divided into 30 columns, of which 3 were reserved 
for fractions, while the remaining 27 were divided into groups 
with 3 columns in each. In every group the columns were 
marked respectively by the letters C {centum), D {decern), 
and S {singularis) or M {monas). Bernelinus gives the nine 
numerals used, which are the apices of Boethius, and then 
remarks that the Greek letters may be used in their place. [3] 


By the use of these columns any number can be written 
without introducing a zero, and all operations in arithmetic 
can be performed in the same way as we execute ours without 
the columns, but with the symbol for zero. Indeed, the 
methods of adding, subtracting, and multiplying in vogue 
among the abacists agree substantially with those of to-day. 
But in a division there is very great difference. The early 
rules for division appear to have been framed to satisfy the 
following three conditions: (1) The use of the multiplication 
table shall be restricted as far as possible; at least, it shall 
never be required to multiply mentally a figure of two digits 
by another of one digit. (2) Subtractions shall be avoided 
as much as possible and replaced by additions. (3) The 
operation shall proceed in a purely mechanical way, without 
requiring trials. [7] That it should be necessary to make such 
conditions seems strange to us; but it must be remembered 
that the monks of the Middle Ages did not attend school 
during childhood and learn the multiplication table while 
the memory was fresh. Gerbert's rules for division are the 
oldest extant. They are so brief as to be very obscure to the 
uninitiated. They were probably intended simply to aid the 
memory by calling to mind the successive steps in the work. In 
later manuscripts they are stated more fully. In dividing any 
number by another of one digit, say 668 by 6, the divisor was 
first increased to 10 by adding 4. The process is exhibited in 
the adjoining figure. [3] As it continues, we must imagine the 
digits which are crossed out, to be erased and then replaced 
by the ones beneath. It is as follows: 600 4- 10 = 60, but, to 



rectify the error, 4 x 60, or 240, must be added; 200 4 10 = 20, 

but 4 x 20, or 80, must be added. We now write for 60 + 40 + 80, 

its sum 180, and continue thus: 100 4 10 = 10; 

the correction necessary is 4 x 10, or 40, which, 

added to 80, gives 120. Now 100 4- 10 = 10, 

and the correction 4 x 10, together with the 20, 

gives 60. Proceeding as before, 60 4 10 = 6; 

the correction is 4 x 6 = 24. Now 20 4 10 = 2, 

the correction being 4x2 = 8. In the column 

of units we have now 8 + 4 + 8, or 20. As 

before, 20 4 10 = 2; the correction is 2 x 4 = 8, 

which is not divisible by 10, but only by 6, 

giving the quotient 1 and the remainder 2. 

All the partial quotients taken together give 

60 + 20 + 10+10 + 6 + 2 + 2 + 1 = 111, and the 

remainder 2. 

Similar but more complicated, is the process 
when the divisor contains two or more digits. 
Were the divisor 27, then the next higher 
multiple of 10, or 30, would be taken for the 
divisor, but corrections would be required for 
the 3. He who has the patience to carry such 
a division through to the end, will understand 
why it has been said of Gerbert that "Regulas dedit, quae a 
sudantibus abacistis vix intelliguntur." He will also perceive 
why the Arabic method of division, when first introduced, 
was called the divisio aurea, but the one on the abacus, the 
divisio ferrea. 































In his book on the abacus, Bernelinus devotes a chapter to 
fractions. These are, of course, the duodecimals, first used 
by the Romans. For want of a suitable notation, calculation 
with them was exceedingly difficult. It would be so even to us, 
were we accustomed, like the early abacists, to express them, 
not by a numerator or denominator, but by the application of 
names, such as uncia for j%, quincunx for ^, dodrans for ^. 

In the tenth century, Gerbert was the central figure among 
the learned. In his time the Occident came into secure 
possession of all mathematical knowledge of the Romans. 
During the eleventh century it was studied assiduously. 
Though numerous works were written on arithmetic and 
geometry, mathematical knowledge in the Occident was still 
very insignificant. Scanty indeed were the mathematical 
treasures obtained from Roman sources. 

Translation of Arabic Manuscripts. 

By his great erudition and phenomenal activity, Gerbert 
infused new life into the study not only of mathematics, but 
also of philosophy. Pupils from France, Germany, and Italy 
gathered at Rheims to enjoy his instruction. When they 
themselves became teachers, they taught of course not only 
the use of the abacus and geometry, but also what they had 
learned of the philosophy of Aristotle. His philosophy was 
known, at first, only through the writings of Boethius. But the 
growing enthusiasm for it created a demand for his complete 
works. Greek texts were wanting. But the Latins heard 


that the Arabs, too, were great admirers of Peripatetism, 
and that they possessed translations of Aristotle's works 
and commentaries thereon. This led them finally to search 
for and translate Arabic manuscripts. During this search, 
mathematical works also came to their notice, and were 
translated into Latin. Though some few unimportant works 
may have been translated earlier, yet the period of greatest 
activity began about 1100. The zeal displayed in acquiring 
the Mohammedan treasures of knowledge excelled even that 
of the Arabs themselves, when, in the eighth century, they 
plundered the rich coffers of Greek and Hindoo science. 

Among the earliest scholars engaged in translating man- 
uscripts into Latin was Athelard of Bath. The period of 
his activity is the first quarter of the twelfth century. He 
travelled extensively in Asia Minor, Egypt, and Spain, and 
braved a thousand perils, that he might acquire the language 
and science of the Mohammedans. He made the earliest 
translations, from the Arabic, of Euclid's Elements and of 
the astronomical tables of Mohammed ben Musa Hovarezmi. 
In 1857, a manuscript was found in the library at Cambridge, 
which proved to be the arithmetic by Mohammed ben Musa in 
Latin. This translation also is very probably due to Athelard. 

At about the same time flourished Plato of Tivoli or Plato 
Tiburtinus. He effected a translation of the astronomy of Al 
Battani and of the Sphterica of Theodosius. Through the 
former, the term sinus was introduced into trigonometry. 

About the middle of the twelfth century there was a group 
of Christian scholars busily at work at Toledo, under the 


leadership of Raymond, then archbishop of Toledo. Among 
those who worked under his direction, John of Seville was 
most prominent. He translated works chiefly on Aristotelian 
philosophy. Of importance to us is a liber algorismi, compiled 
by him from Arabic authors. On comparing works like 
this with those of the abacists, we notice at once the most 
striking difference, which shows that the two parties drew from 
independent sources. It is argued by some that Gerbert got 
his apices and his arithmetical knowledge, not from Boethius, 
but from the Arabs in Spain, and that part or the whole of 
the geometry of Boethius is a forgery, dating from the time of 
Gerbert. If this were the case, then the writings of Gerbert 
would betray Arabic sources, as do those of John of Seville. 
But no points of resemblance are found. Gerbert could not 
have learned from the Arabs the use of the abacus, because 
all evidence we have goes to show that they did not employ it. 
Nor is it probable that he borrowed from the Arabs the apices, 
because they were never used in Europe except on the abacus. 
In illustrating an example in division, mathematicians of 
the tenth and eleventh centuries state an example in Roman 
numerals, then draw an abacus and insert in it the necessary 
numbers with the apices. Hence it seems probable that 
the abacus and apices were borrowed from the same source. 
The contrast between authors like John of Seville, drawing 
from Arabic works, and the abacists, consists in this, that, 
unlike the latter, the former mention the Hindoos, use the 
term algorism, calculate with the zero, and do not employ 
the abacus. The former teach the extraction of roots, the 


abacists do not; they teach the sexagesimal fractions used by 
the Arabs, while the abacists employ the duodecimals of the 
Romans. [3] 

A little later than John of Seville flourished Gerard of 
Cremona in Lombardy. Being desirous to gain possession 
of the Almagest, he went to Toledo, and there, in 1175, 
translated this great work of Ptolemy. Inspired by the 
richness of Mohammedan literature, he gave himself up to its 
study. He translated into Latin over 70 Arabic works. Of 
mathematical treatises, there were among these, besides the 
Almagest, the 15 books of Euclid, the Sphterica of Theodosius, 
a work of Menelaus, the algebra of Mohammed ben Musa 
Hovarezmi, the astronomy of Dshabir ben Aflah, and others 
less important. 

In the thirteenth century, the zeal for the acquisition of 
Arabic learning continued. Foremost among the patrons 
of science at this time ranked Emperor Frederick II. of 
Hohenstaufen (died 1250). Through frequent contact with 
Mohammedan scholars, he became familiar with Arabic 
science. He employed a number of scholars in translating 
Arabic manuscripts, and it was through him that we came 
in possession of a new translation of the Almagest. Another 
royal head deserving mention as a zealous promoter of Arabic 
science was Alfonso X. of Castile (died 1284). He gathered 
around him a number of Jewish and Christian scholars, who 
translated and compiled astronomical works from Arabic 
sources. Rabbi Zag and Iehuda ben Mose Cohen were 
the most prominent among them. Astronomical tables 


prepared by these two Jews spread rapidly in the Occident, 
and constituted the basis of all astronomical calculation till 
the sixteenth century [7] The number of scholars who aided 
in transplanting Arabic science upon Christian soil was large. 
But we mention only one more. Giovanni Campano of 
Novara (about 1260) brought out a new translation of Euclid, 
which drove the earlier ones from the field, and which formed 
the basis of the printed editions. [7] 

At the close of the twelfth century, the Occident was in 
possession of the so-called Arabic notation. The Hindoo 
methods of calculation began to supersede the cumbrous 
methods inherited from Rome. Algebra, with its rules 
for solving linear and quadratic equations, had been made 
accessible to the Latins. The geometry of Euclid, the Sphcerica 
of Theodosius, the astronomy of Ptolemy, and other works 
were now accessible in the Latin tongue. Thus a great amount 
of new scientific material had come into the hands of the 
Christians. The talent necessary to digest this heterogeneous 
mass of knowledge was not wanting. The figure of Leonardo 
of Pisa adorns the vestibule of the thirteenth century. 

It is important to notice that no work either on mathematics 
or astronomy was translated directly from the Greek previous 
to the fifteenth century. 

The First Awakening and its Sequel. 

Thus far, France and the British Isles have been the 
headquarters of mathematics in Christian Europe. But at the 


beginning of the thirteenth century the talent and activity 
of one man was sufficient to assign the mathematical science 
a new home in Italy This man was not a monk, like Bede, 
Alcuin, or Gerbert, but a merchant, who in the midst of 
business pursuits found time for scientific study Leonardo 
of Pisa is the man to whom we owe the first renaissance of 
mathematics on Christian soil. He is also called Fibonacci, 
i.e. son of Bonaccio. His father was secretary at one of the 
numerous factories erected on the south and east coast of 
the Mediterranean by the enterprising merchants of Pisa. He 
made Leonardo, when a boy, learn the use of the abacus. The 
boy acquired a strong taste for mathematics, and, in later 
years, during his extensive business travels in Egypt, Syria, 
Greece, and Sicily, collected from the various peoples all the 
knowledge he could get on this subject. Of all the methods 
of calculation, he found the Hindoo to be unquestionably the 
best. Returning to Pisa, he published, in 1202, his great 
work, the Liber Abaci. A revised edition of this appeared 
in 1228. This work contains about all the knowledge the Arabs 
possessed in arithmetic and algebra, and treats the subject 
in a free and independent way. This, together with the other 
books of Leonardo, shows that he was not merely a compiler, 
or, like other writers of the Middle Ages, a slavish imitator of 
the form in which the subject had been previously presented, 
but that he was an original worker of exceptional power. 

He was the first great mathematician to advocate the 
adoption of the "Arabic notation." The calculation with the 
zero was the portion of Arabic mathematics earliest adopted 


by the Christians. The minds of men had been prepared for 
the reception of this by the use of the abacus and the apices. 
The reckoning with columns was gradually abandoned, and 
the very word abacus changed its meaning and became a 
synonym for algorism. For the zero, the Latins adopted the 
name zephirum, from the Arabic sifr (si/ra=empty); hence 
our English word cipher. The new notation was accepted 
readily by the enlightened masses, but, at first, rejected by 
the learned circles. The merchants of Italy used it as early 
as the thirteenth century, while the monks in the monasteries 
adhered to the old forms. In 1299, nearly 100 years after 
the publication of Leonardo's Liber Abaci, the Florentine 
merchants were forbidden the use of the Arabic numerals 
in book-keeping, and ordered either to employ the Roman 
numerals or to write the numeral adjectives out in full. In 
the fifteenth century the abacus with its counters ceased to 
be used in Spain and Italy. In France it was used later, 
and it did not disappear in England and Germany before the 
middle of the seventeenth century. [22] Thus, in the Winter's 
Tale (iv. 3), Shakespeare lets the clown be embarrassed by 
a problem which he could not do without counters. Iago 
(in Othello, i. 1) expresses his contempt for Michael Cassio, 
"forsooth a great mathematician," by calling him a "counter- 
caster." So general, indeed, says Peacock, appears to have 
been the practice of this species of arithmetic, that its rules and 
principles form an essential part of the arithmetical treatises 
of that day. The real fact seems to be that the old methods 
were used long after the Hindoo numerals were in common 


and general use. With such dogged persistency does man 
cling to the old! 

The Liber Abaci was, for centuries, the storehouse from 
which authors got material for works on arithmetic and 
algebra. In it are set forth the most perfect methods of 
calculation with integers and fractions, known at that time; 
the square and cube root are explained; equations of the first 
and second degree leading to problems, either determinate or 
indeterminate, are solved by the methods of 'single' or 'double 
position,' and also by real algebra. The book contains a large 
number of problems. The following was proposed to Leonardo 
of Pisa by a magister in Constantinople, as a difficult problem: 
If A gets from B 7 denare, then A's sum is five- fold B's; if B 
gets from A 5 denare, then B's sum is seven- fold A's. How 
much has each? The Liber Abaci contains another problem, 
which is of historical interest, because it was given with some 
variations by Ahmes, 3000 years earlier: 7 old women go to 
Rome; each woman has 7 mules, each mule carries 7 sacks, 
each sack contains 7 loaves, with each loaf are 7 knives, each 
knife is put up in 7 sheaths. What is the sum total of all 
named? Ans. 137,256. [3] 

In 1220, Leonardo of Pisa published his Practica Ge- 
ometri(E, which contains all the knowledge of geometry and 
trigonometry transmitted to him. The writings of Euclid 
and of some other Greek masters were known to him, either 
from Arabic manuscripts directly or from the translations 
made by his countrymen, Gerard of Cremona and Plato of 
Tivoli. Leonardo's Geometry contains an elegant geometrical 


demonstration of Heron's formula for the area of a triangle, as 
a function of its three sides. Leonardo treats the rich material 
before him with skill and Euclidean rigour. 

Of still greater interest than the preceding works are those 
containing Fibonacci's original investigations. We must here 
preface that after the publication of the Liber Abaci, Leonardo 
was presented by the astronomer Dominicus to Emperor 
Frederick II. of Hohenstaufen. On that occasion, John of 
Palermo, an imperial notary, proposed several problems, 
which Leonardo solved promptly. The first problem was to 
find a number x, such that x 2 + 5 and x 2 — 5 are each square 
numbers. The answer is x = 3^; for (3^) 2 + 5 = (4^) 2 , 
(3j7j) 2 — 5 = (2^) 2 . His masterly solution of this is given 
in his liber quadratorum, a copy of which work was sent by 
him to Frederick II. The problem was not original with John 
of Palermo, since the Arabs had already solved similar ones. 
Some parts of Leonardo's solution may have been borrowed 
from the Arabs, but the method which he employed of building 
squares by the summation of odd numbers is original with 

The second problem proposed to Leonardo at the famous 
scientific tournament which accompanied the presentation of 
this celebrated algebraist to that great patron of learning, 
Emperor Frederick II., was the solving of the equation 
x 3 + 2x 2 + 10a: = 20. As yet cubic equations had not been 
solved algebraically. Instead of brooding stubbornly over 
this knotty problem, and after many failures still entertaining 
new hopes of success, he changed his method of inquiry and 


showed by clear and rigorous demonstration that the roots 

of this equation could not be represented by the Euclidean 

irrational quantities, or, in other words, that they could not 

be constructed with the ruler and compass only He contented 

himself with finding a very close approximation to the required 

root. His work on this cubic is found in the Flos, together with 

the solution of the following third problem given him by John 

of Palermo: Three men possess in common an unknown sum 

of money t; the share of the first is -; that of the second, -; 
f 2 3 

that of the third, -. Desirous of depositing the sum at a safer 

place, each takes at hazard a certain amount; the first takes x, 

X V 

but deposits only -; the second carries y, but deposits only -; 
the third takes z, and deposits -. Of the amount deposited 
each one must receive exactly |, in order to possess his share 
of the whole sum. Find x, y, z. Leonardo shows the problem 
to be indeterminate. Assuming 7 for the sum drawn by each 
from the deposit, he finds t = 47, x = 33, y = 13, z = 1. 

One would have thought that after so brilliant a beginning, 
the sciences transplanted from Mohammedan to Christian 
soil would have enjoyed a steady and vigorous development. 
But this was not the case. During the fourteenth and fifteenth 
centuries, the mathematical science was almost stationary. 
Long wars absorbed the energies of the people and thereby 
kept back the growth of the sciences. The death of Frederick II. 
in 1254 was followed by a period of confusion in Germany. The 
German emperors and the popes were continually quarrelling, 
and Italy was inevitably drawn into the struggles between 
the Guelphs and the Ghibellines. France and England were 


engaged in the Hundred Years' War (1338-1453). Then 
followed in England the Wars of the Roses. The growth of 
science was retarded not only by war, but also by the injurious 
influence of scholastic philosophy. The intellectual leaders 
of those times quarrelled over subtle subjects in metaphysics 
and theology. Frivolous questions, such as "How many angels 
can stand on the point of a needle?" were discussed with great 
interest. Indistinctness and confusion of ideas characterised 
the reasoning during this period. Among the mathematical 
productions of the Middle Ages, the works of Leonardo of Pisa 
appear to us like jewels among quarry-rubbish. The writers 
on mathematics during this period were not few in number, 
but their scientific efforts were vitiated by the method of 
scholastic thinking. Though they possessed the Elements of 
Euclid, yet the true nature of a mathematical proof was so 
little understood, that Hankel believes it no exaggeration to 
say that "since Fibonacci, not a single proof, not borrowed 
from Euclid, can be found in the whole literature of these 
ages, which fulfils all necessary conditions." 

The only noticeable advance is a simplification of numerical 
operations and a more extended application of them. Among 
the Italians are evidences of an early maturity of arithmetic. 
Peacock [22] says: The Tuscans generally, and the Florentines 
in particular, whose city was the cradle of the literature and 
arts of the thirteenth and fourteenth centuries, were celebrated 
for their knowledge of arithmetic and book-keeping, which 
were so necessary for their extensive commerce; the Italians 
were in familiar possession of commercial arithmetic long 


before the other nations of Europe; to them we are indebted 
for the formal introduction into books of arithmetic, under 
distinct heads, of questions in the single and double rule 
of three, loss and gain, fellowship, exchange, simple and 
compound interest, discount, and so on. 

There was also a slow improvement in the algebraic no- 
tation. The Hindoo algebra possessed a tolerable symbolic 
notation, which was, however, completely ignored by the 
Mohammedans. In this respect, Arabic algebra approached 
much more closely to that of Diophantus, which can scarcely 
be said to employ symbols in a systematic way. Leonardo of 
Pisa possessed no algebraic symbolism. Like the Arabs, he 
expressed the relations of magnitudes to each other by lines or 
in words. But in the mathematical writings of the monk Luca 
Pacioli (also called Lucas de Burgo sepulchri) symbols began 
to appear. They consisted merely in abbreviations of Italian 
words, such as p for piu (more), m for meno (less), co for cosa 
(the thing or unknown quantity). "Our present notation has 
arisen by almost insensible degrees as convenience suggested 
different marks of abbreviation to different authors; and that 
perfect symbolic language which addresses itself solely to the 
eye, and enables us to take in at a glance the most complicated 
relations of quantity, is the result of a large series of small 
improvements." [23] 

We shall now mention a few authors who lived during the 
thirteenth and fourteenth and the first half of the fifteenth 
centuries. About the time of Leonardo of Pisa (1200 A.D.), 
lived the German monk Jordanus Nemorarius, who wrote 


a once famous work on the properties of numbers (1496), 
modelled after the arithmetic of Boethius. The most trifling 
numeral properties are treated with nauseating pedantry 
and prolixity. A practical arithmetic based on the Hindoo 
notation was also written by him. John Halifax (Sacro 
Bosco, died 1256) taught in Paris and made an extract from 
the Almagest containing only the most elementary parts of 
that work. This extract was for nearly 400 years a work of 
great popularity and standard authority. Other prominent 
writers are Albertus Magnus and George Purbach in 
Germany, and Roger Bacon in England. It appears that 
here and there some of our modern ideas were anticipated 
by writers of the Middle Ages. Thus, Nicole Oresme, a 
bishop in Normandy (died 1382), first conceived a notation 
of fractional powers, afterwards re-discovered by Stevinus, 
and gave rules for operating with them. His notation 
was totally different from ours. Thomas Bradwardine, 
archbishop of Canterbury, studied star-polygons, — a subject 
which has recently received renewed attention. The first 
appearance of such polygons was with Pythagoras and his 
school. We next meet with such polygons in the geometry of 
Boethius and also in the translation of Euclid from the Arabic 
by Athelard of Bath. Bradwardine's philosophic writings 
contain discussions on the infinite and the infinitesimal — 
subjects never since lost sight of. To England falls the 
honour of having produced the earliest European writers on 
trigonometry. The writings of Bradwardine, of Richard of 
Wallingford, and John Maudith, both professors at Oxford, 


and of Simon Bredon of Winchecombe, contain trigonometry 
drawn from Arabic sources. 

The works of the Greek monk Maximus Planudes, who 
lived in the first half of the fourteenth century, are of interest 
only as showing that the Hindoo numerals were then known in 
Greece. A writer belonging, like Planudes, to the Byzantine 
school, was Moschopulus, who lived in Constantinople in 
the early part of the fifteenth century. To him appears to be 
due the introduction into Europe of magic squares. He wrote 
a treatise on this subject. Magic squares were known to the 
Arabs, and perhaps to the Hindoos. Mediaeval astrologers 
and physicians believed them to possess mystical properties 
and to be a charm against plague, when engraved on silver 

In 1494 was printed the Summa de Arithmetica, Geometria, 
Proportione et Proportionalita, written by the Tuscan monk 
Lucas Pacioli, who, as we remarked, first introduced symbols 
in algebra. This contains all the knowledge of his day 
on arithmetic, algebra, and trigonometry, and is the first 
comprehensive work which appeared after the Liber Abaci of 
Fibonacci. It contains little of importance which cannot be 
found in Fibonacci's great work, published three centuries 
earlier. [1] 

Perhaps the greatest result of the influx of Arabic learning 
was the establishment of universities. What was their attitude 
toward mathematics? The University of Paris, so famous at 
the beginning of the twelfth century under the teachings of 
Abelard, paid but little attention to this science during the 


Middle Ages. Geometry was neglected, and Aristotle's logic 
was the favourite study. In 1336, a rule was introduced that 
no student should take a degree without attending lectures on 
mathematics, and from a commentary on the first six books of 
Euclid, dated 1536, it appears that candidates for the degree 
of A.M. had to give an oath that they had attended lectures 
on these books. [7] Examinations, when held at all, probably 
did not extend beyond the first book, as is shown by the 
nickname "magister matheseos," applied to the Theorem of 
Pythagoras, the last in the first book. More attention was 
paid to mathematics at the University of Prague, founded 
1384. For the Baccalaureate degree, students were required 
to take lectures on Sacro Bosco's famous work on astronomy. 
Of candidates for the A.M. were required not only the six 
books of Euclid, but an additional knowledge of applied 
mathematics. Lectures were given on the Almagest. At the 
University of Leipzig, the daughter of Prague, and at Cologne, 
less work was required, and, as late as the sixteenth century, 
the same requirements were made at these as at Prague in 
the fourteenth. The universities of Bologna, Padua, Pisa, 
occupied similar positions to the ones in Germany, only that 
purely astrological lectures were given in place of lectures 
on the Almagest. At Oxford, in the middle of the fifteenth 
century, the first two books of Euclid were read. [6] 

Thus it will be seen that the study of mathematics was 
maintained at the universities only in a half-hearted manner. 
No great mathematician and teacher appeared, to inspire the 
students. The best energies of the schoolmen were expended 


upon the stupid subtleties of their philosophy. The genius of 
Leonardo of Pisa left no permanent impress upon the age, and 
another Renaissance of mathematics was wanted. 


We find it convenient to choose the time of the capture 
of Constantinople by the Turks as the date at which the 
Middle Ages ended and Modern Times began. In 1453, the 
Turks battered the walls of this celebrated metropolis with 
cannon, and finally captured the city; the Byzantine Empire 
fell, to rise no more. Calamitous as was this event to the 
East, it acted favourably upon the progress of learning in 
the West. A great number of learned Greeks fled into Italy, 
bringing with them precious manuscripts of Greek literature. 
This contributed vastly to the reviving of classic learning. 
Up to this time, Greek masters were known only through 
the often very corrupt Arabic manuscripts, but now they 
began to be studied from original sources and in their own 
language. The first English translation of Euclid was made 
in 1570 from the Greek by Sir Henry Billingsley, assisted by 
John Dee. [29] About the middle of the fifteenth century, 
printing was invented; books became cheap and plentiful; 
the printing-press transformed Europe into an audience- 
room. Near the close of the fifteenth century, America was 
discovered, and, soon after, the earth was circumnavigated. 
The pulse and pace of the world began to quicken. Men's 
minds became less servile; they became clearer and stronger. 
The indistinctness of thought, which was the characteristic 
feature of mediaeval learning, began to be remedied chiefly by 
the steady cultivation of Pure Mathematics and Astronomy. 



Dogmatism was attacked; there arose a long struggle with 
the authority of the Church and the established schools of 
philosophy. The Copernican System was set up in opposition 
to the time-honoured Ptolemaic System. The long and eager 
contest between the two culminated in a crisis at the time 
of Galileo, and resulted in the victory of the new system. 
Thus, by slow degrees, the minds of men were cut adrift from 
their old scholastic moorings and sent forth on the wide sea 
of scientific inquiry, to discover new islands and continents of 


With the sixteenth century began a period of increased 
intellectual activity. The human mind made a vast effort 
to achieve its freedom. Attempts at its emancipation from 
Church authority had been made before, but they were 
stifled and rendered abortive. The first great and successful 
revolt against ecclesiastical authority was made in Germany. 
The new desire for judging freely and independently in 
matters of religion was preceded and accompanied by a 
growing spirit of scientific inquiry. Thus it was that, for 
a time, Germany led the van in science. She produced 
Regiomontanus, Copernicus, Rhosticus, Kepler, and Tycho 
Brake, at a period when France and England had, as yet, 
brought forth hardly any great scientific thinkers. This 
remarkable scientific productiveness was no doubt due, to 
a great extent, to the commercial prosperity of Germany. 


Material prosperity is an essential condition for the progress 
of knowledge. As long as every individual is obliged to collect 
the necessaries for his subsistence, there can be no leisure 
for higher pursuits. At this time, Germany had accumulated 
considerable wealth. The Hanseatic League commanded 
the trade of the North. Close commercial relations existed 
between Germany and Italy. Italy, too, excelled in commercial 
activity and enterprise. We need only mention Venice, whose 
glory began with the crusades, and Florence, with her bankers 
and her manufacturers of silk and wool. These two cities 
became great intellectual centres. Thus, Italy, too, produced 
men in art, literature, and science, who shone forth in fullest 
splendour. In fact, Italy was the fatherland of what is termed 
the Renaissance. 

For the first great contributions to the mathematical 
sciences we must, therefore, look to Italy and Germany. In 
Italy brilliant accessions were made to algebra, in Germany 
to astronomy and trigonometry. 

On the threshold of this new era we meet in Germany with 
the figure of John Mueller, more generally called Regiomon- 
tanus (1436-1476). Chiefly to him we owe the revival of 
trigonometry. He studied astronomy and trigonometry at 
Vienna under the celebrated George Purbach. The latter 
perceived that the existing Latin translations of the Almagest 
were full of errors, and that Arabic authors had not remained 
true to the Greek original. Purbach therefore began to make 
a translation directly from the Greek. But he did not live 
to finish it. His work was continued by Regiomontanus, who 


went beyond his master. Regiomontanus learned the Greek 
language from Cardinal Bessarion, whom he followed to Italy, 
where he remained eight years collecting manuscripts from 
Greeks who had fled thither from the Turks. In addition to 
the translation of and the commentary on the Almagest, he 
prepared translations of the Conies of Apollonius, of Archi- 
medes, and of the mechanical works of Heron. Regiomontanus 
and Purbach adopted the Hindoo sine in place of the Greek 
chord of double the arc. The Greeks and afterwards the Arabs 
divided the radius into 60 equal parts, and each of these again 
into 60 smaller ones. The Hindoos expressed the length of the 
radius by parts of the circumference, saying that of the 21, 600 
equal divisions of the latter, it took 3438 to measure the ra- 
dius. Regiomontanus, to secure greater precision, constructed 
one table of sines on a radius divided into 600,000 parts, 
and another on a radius divided decimally into 10, 000, 000 
divisions. He emphasised the use of the tangent in trigono- 
metry. Following out some ideas of his master, he calculated a 
table of tangents. German mathematicians were not the first 
Europeans to use this function. In England it was known a 
century earlier to Bradwardine, who speaks of tangent (umbra 
recta) and cotangent (umbra versa), and to John Maudith. 
Regiomontanus was the author of an arithmetic and also of 
a complete treatise on trigonometry, containing solutions of 
both plane and spherical triangles. The form which he gave 
to trigonometry has been retained, in its main features, to the 
present day. 

Regiomontanus ranks among the greatest men that Ger- 


many has ever produced. His complete mastery of astronomy 
and mathematics, and his enthusiasm for them, were of 
far-reaching influence throughout Germany. So great was 
his reputation, that Pope Sixtus IV. called him to Italy to 
improve the calendar. Regiomontanus left his beloved city of 
Nurnberg for Rome, where he died in the following year. 

After the time of Purbach and Regiomontanus, trigono- 
metry and especially the calculation of tables continued to 
occupy German scholars. More refined astronomical in- 
struments were made, which gave observations of greater 
precision; but these would have been useless without trigono- 
metrical tables of corresponding accuracy. Of the several 
tables calculated, that by Georg Joachim of Feldkirch in 
Tyrol, generally called Rhaeticus, deserves special mention. 
He calculated a table of sines with the radius =10, 000, 000, 000 
and from 10" to 10"; and, later on, another with the radius 
= 1,000,000,000,000,000, and proceeding from 10" to 10". 
He began also the construction of tables of tangents and 
secants, to be carried to the same degree of accuracy; but 
he died before finishing them. For twelve years he had had 
in continual employment several calculators. The work was 
completed by his pupil, Valentine Otho, in 1596. This was 
indeed a gigantic work, — a monument of German diligence 
and indefatigable perseverance. The tables were republished 
in 1613 by Pitiscus, who spared no pains to free them of 
errors. Astronomical tables of so great a degree of accu- 
racy had never been dreamed of by the Greeks, Hindoos, or 
Arabs. That Rhaeticus was not a ready calculator only, is 


indicated by his views on trigonometrical lines. Up to his 
time, the trigonometric functions had been considered always 
with relation to the arc; he was the first to construct the right 
triangle and to make them depend directly upon its angles. 
It was from the right triangle that Rhaeticus got his idea of 
calculating the hypotenuse; i. e. he was the first to plan a table 
of secants. Good work in trigonometry was done also by Vieta 
and Romanus. 

We shall now leave the subject of trigonometry to witness 
the progress in the solution of algebraical equations. To do 
so, we must quit Germany for Italy. The first comprehensive 
algebra printed was that of Lucas Pacioli. He closes his book 
by saying that the solution of the equations x 3, + mx = n, 
x s + n = mx is as impossible at the present state of science as 
the quadrature of the circle. This remark doubtless stimulated 
thought. The first step in the algebraic solution of cubics was 
taken by ScipioFerro (died 1526), a professor of mathematics 
at Bologna, who solved the equation x 3 + mx = n. Nothing 
more is known of his discovery than that he imparted it to 
his pupil, Floridas, in 1505. It was the practice in those days 
and for two centuries afterwards to keep discoveries secret, 
in order to secure by that means an advantage over rivals 
by proposing problems beyond their reach. This practice 
gave rise to numberless disputes regarding the priority of 
inventions. A second solution of cubics was given by Nicolo 
of Brescia (1506(?)-1557). When a boy of six, Nicolo was 
so badly cut by a French soldier that he never again gained 
the free use of his tongue. Hence he was called Tartaglia, 


i.e. the stammerer. His widowed mother being too poor to 
pay his tuition in school, he learned to read and picked up 
a knowledge of Latin, Greek, and mathematics by himself. 
Possessing a mind of extraordinary power, he was able to 
appear as teacher of mathematics at an early age. In 1530, 
one Colla proposed him several problems, one leading to the 
equation x 3 + px 2 = q. Tartaglia found an imperfect method 
for solving this, but kept it secret. He spoke about his secret in 
public and thus caused Ferro's pupil, Floridas, to proclaim his 
own knowledge of the form x 3 + mx = n. Tartaglia, believing 
him to be a mediocrist and braggart, challenged him to a 
public discussion, to take place on the 22d of February, 1535. 
Hearing, meanwhile, that his rival had gotten the method 
from a deceased master, and fearing that he would be beaten 
in the contest, Tartaglia put in all the zeal, industry, and 
skill to find the rule for the equations, and he succeeded in it 
ten days before the appointed date, as he himself modestly 
says. [7] The most difficult step was, no doubt, the passing 
from quadratic irrationals, used in operating from time of 
old, to cubic irrationals. Placing x = \/i — tyu, Tartaglia 
perceived that the irrationals disappeared from the equation 
x 3 + mx = n, making n = t — u. But this last equality, together 
with (j>m) 3 = tu, gives at once 

n\3 /m\3 n /n\ 2 /m\ 3 n 

2) + {j) + 2' u = \l(2) + (V "2- 

This is Tartaglia's solution of x 3 + mx = n. On the 13th of 
February, he found a similar solution for x 3 = mx + n. The 
contest began on the 22d. Each contestant proposed thirty 


problems. The one who could solve the greatest number 
within fifty days should be the victor. Tartaglia solved the 
thirty problems proposed by Floridas in two hours; Floridas 
could not solve any of Tartaglia's. From now on, Tartaglia 
studied cubic equations with a will. In 1541 he discovered a 
general solution for the cubic x 3 ± px 2 = ±q, by transforming 
it into the form x ± mx = ±n. The news of Tartaglia's 
victory spread all over Italy. Tartaglia was entreated to 
make known his method, but he declined to do so, saying 
that after his completion of the translation from the Greek 
of Euclid and Archimedes, he would publish a large algebra 
containing his method. But a scholar from Milan, named 
Hieronimo Cardano (1501-1576), after many solicitations, 
and after giving the most solemn and sacred promises of 
secrecy, succeeded in obtaining from Tartaglia a knowledge of 
his rules. 

At this time Cardan was writing his Ars Magna, and he 
knew no better way to crown his work than by inserting the 
much sought for rules for solving cubics. Thus Cardan broke 
his most solemn vows, and published in 1545 in his Ars Magna 
Tartaglia's solution of cubics. Tartaglia became desperate. 
His most cherished hope, of giving to the world an immortal 
work which should be the monument of his deep learning and 
power for original research, was suddenly destroyed; for the 
crown intended for his work had been snatched away. His 
first step was to write a history of his invention; but, to 
completely annihilate his enemies, he challenged Cardan and 
his pupil Lodovico Ferrari to a contest: each party should 


propose thirty-one questions to be solved by the other within 
fifteen days. Tartaglia solved most questions in seven days, 
but the other party did not send in their solution before the 
expiration of the fifth month; moreover, all their solutions 
except one were wrong. A replication and a rejoinder followed. 
Endless were the problems proposed and solved on both sides. 
The dispute produced much chagrin and heart-burnings to 
the parties, and to Tartaglia especially, who met with many 
other disappointments. After having recovered himself again, 
Tartaglia began, in 1556, the publication of the work which he 
had had in his mind for so long; but he died before he reached 
the consideration of cubic equations. Thus the fondest wish 
of his life remained unfulfilled; the man to whom we owe the 
greatest contribution to algebra made in the sixteenth century 
was forgotten, and his method came to be regarded as the 
discovery of Cardan and to be called Cardan's solution. 

Remarkable is the great interest that the solution of cubics 
excited throughout Italy. It is but natural that after this 
great conquest mathematicians should attack biquadratic 
equations. As in the case of cubics, so here, the first impulse 
was given by Colla, who, in 1540, proposed for solution 
the equation a; 4 + 6a: 2 + 36 = 60x. To be sure, Cardan had 
studied particular cases as early as 1539. Thus he solved 
the equation 13x 2 = x 4 + 2a; 3 + 2x + 1 by a process similar 
to that employed by Diophantus and the Hindoos; namely, 
by adding to both sides 3x 2 and thereby rendering both 
numbers complete squares. But Cardan failed to find a 
general solution; it remained for his pupil Ferrari to prop 


the reputation of his master by the brilliant discovery of the 
general solution of biquadratic equations. Ferrari reduced 
Colla's equation to the form (x 2 + 6) 2 = 60a; + 6x 2 . In order 
to give also the right member the form of a complete square 
he added to both members the expression 2(x 2 + 6)y + y 2 , 
containing a new unknown quantity y. This gave him 
(x 2 + 6 + y) 2 = (6 + 2y)x 2 + 60x + (12y + y 2 ). The condition 
that the right member be a complete square is expressed by the 

cubic equation (2y + 6)(12y + y 2 ) = 900. Extracting the square 

. , , . , . , 9 , 900 

root of the biquadratic, he got x +6 + y = x\/2y + 6 + - 

Solving the cubic for y and substituting, it remained only to 
determine x from the resulting quadratic. Ferrari pursued a 
similar method with other numerical biquadratic equations. [7] 
Cardan had the pleasure of publishing this discovery in his 
Ars Magna in 1545. Ferrari's solution is sometimes ascribed 
to Bombelli, but he is no more the discoverer of it than Cardan 
is of the solution called by his name. 

To Cardan algebra is much indebted. In his Ars Magna 
he takes notice of negative roots of an equation, calling them 
fictitious, while the positive roots are called real. Imaginary 
roots he does not consider; cases where they appear he 
calls impossible. Cardan also observed the difficulty in the 
irreducible case in the cubics, which, like the quadrature of the 
circle, has since "so much tormented the perverse ingenuity of 
mathematicians." But he did not understand its nature. It 
remained for Raphael Bombelli of Bologna, who published 
in 1572 an algebra of great merit, to point out the reality of 
the apparently imaginary expression which the root assumes, 


and thus to lay the foundation of a more intimate knowledge 
of imaginary quantities. 

After this brilliant success in solving equations of the third 
and fourth degrees, there was probably no one who doubted, 
that with aid of irrationals of higher degrees, the solution 
of equations of any degree whatever could be found. But 
all attempts at the algebraic solution of the quintic were 
fruitless, and, finally, Abel demonstrated that all hopes of 
finding algebraic solutions to equations of higher than the 
fourth degree were purely Utopian. 

Since no solution by radicals of equations of higher degrees 
could be found, there remained nothing else to be done than 
the devising of rules by which at least the numerical values of 
the roots could be ascertained. Cardan applied the Hindoo 
rule of "false position" (called by him regula aurea) to the 
cubic, but this mode of approximating was exceedingly rough. 
An incomparably better method was invented by Franciscus 
Vieta, a French mathematician, whose transcendent genius 
enriched mathematics with several important innovations. 
Taking the equation f(x) = Q, wherein f(x) is a polynomial 
containing different powers of x, with numerical coefficients, 
and Q is a given number, Vieta first substitutes in f(x) a 
known approximate value of the root, and then shows that 
another figure of the root can be obtained by division. A 
repetition of the same process gives the next figure of the 
root, and so on. Thus, in x 1 + 14x = 7929, taking 80 for the 


approximate root, and placing x = 80 + 6, we get 

(80 + b) 2 + 14(80 + b) = 7929, 
or 1746 + b 2 = 409. 

Since 1746 is much greater than 6 2 , we place 1746 = 409, 
and obtain thereby 6=2. Hence the second approximation 
is 82. Put x = 82 + c, then (82 + c) 2 + 14(82 + c) = 7929, or 
178c + c 2 = 57. As before, place 178c = 57, then c = .3, and 
the third approximation gives 82.3. Assuming x = 82.3 + d, 
and substituting, gives 178. 6d + d 2 = 3.51, and 178. 6d = 3.51, 
.•. d = .01; giving for the fourth approximation 82.31. In the 
same way, e = .009, and the value for the root of the given 

equation is 82.319 For this process, Vieta was greatly 

admired by his contemporaries. It was employed by Harriot, 
Oughtred, Pell, and others. Its principle is identical with the 
main principle involved in the methods of approximation of 
Newton and Horner. The only change lies in the arrangement 
of the work. This alteration was made to afford facility and 
security in the process of evolution of the root. 

We pause a moment to sketch the life of Vieta, the most 
eminent French mathematician of the sixteenth century. He 
was born in Poitou in 1540, and died in 1603 at Paris. He 
was employed throughout life in the service of the state, 
under Henry III. and Henry IV. He was, therefore, not a 
mathematician by profession, but his love for the science was 
so great that he remained in his chamber studying, sometimes 
several days in succession, without eating and sleeping more 
than was necessary to sustain himself. So great devotion to 


abstract science is the more remarkable, because he lived at a 
time of incessant political and religious turmoil. During the 
war against Spain, Vieta rendered service to Henry IV. by 
deciphering intercepted letters written in a species of cipher, 
and addressed by the Spanish Court to their governor of 
Netherlands. The Spaniards attributed the discovery of the 
key to magic. 

An ambassador from Netherlands once told Henry IV. 
that France did not possess a single geometer capable of 
solving a problem propounded to geometers by a Belgian 
mathematician, Adrianus Romanus. It was the solution of 
the equation of the forty-fifth degree:— 

45y - 3795y 3 + 95634y 5 + 945y 41 - 45y 43 + y 45 = C. 

Henry IV. called Vieta, who, having already pursued similar 
investigations, saw at once that this awe-inspiring problem 
was simply the equation by which C = 2 sin <f> was expressed in 
terms of y = 2 sin ^<f>; that, since 45 = 3-3-5, it was necessary 
only to divide an angle once into 5 equal parts, and then twice 
into 3, — a division which could be effected by corresponding 
equations of the fifth and third degrees. Brilliant was the 
discovery by Vieta of 23 roots to this equation, instead of 
only one. The reason why he did not find 45 solutions, is 
that the remaining ones involve negative sines, which were 
unintelligible to him. Detailed investigations on the famous 
old problem of the section of an angle into an odd number of 
equal parts, led Vieta to the discovery of a trigonometrical 
solution of Cardan's irreducible case in cubics. He applied the 


equation (2 cos ^<j>) — 3 (2 cos \<t>) = 2cos</> to the solution of 
x 3 — 3a 2 x = a 2 b, when a > r>b, by placing x = 2a cos g0, and 
determining </> from b = 2a cos </>. 

The main principle employed by him in the solution of 
equations is that of reduction. He solves the quadratic by 
making a suitable substitution which will remove the term 
containing x to the first degree. Like Cardan, he reduces the 
general expression of the cubic to the form x 3 + mx + n = 0; 
then, assuming x = (|a — z 2 ) -=- z and substituting, he gets 
z 6 — bz^ — ^a 3 = 0. Putting z 3 = y, he has a quadratic. 
In the solution of biquadratics, Vieta still remains true to 
his principle of reduction. This gives him the well-known 
cubic resolvent. He thus adheres throughout to his favourite 
principle, and thereby introduces into algebra a uniformity of 
method which claims our lively admiration. In Vieta's algebra 
we discover a partial knowledge of the relations existing 
between the coefficients and the roots of an equation. He 
shows that if the coefficient of the second term in an equation 
of the second degree is minus the sum of two numbers whose 
product is the third term, then the two numbers are roots of 
the equation. Vieta rejected all except positive roots; hence 
it was impossible for him to fully perceive the relations in 

The most epoch-making innovation in algebra due to Vieta 
is the denoting of general or indefinite quantities by letters 
of the alphabet. To be sure, Regiomontanus and Stifel in 
Germany, and Cardan in Italy, used letters before him, but 
Vieta extended the idea and first made it an essential part of 


algebra. The new algebra was called by him logistica speciosa 
in distinction to the old logistica numerosa. Vieta's formalism 
differed considerably from that of to-day. The equation 
a 3 + 3a 2 b + 3ab 2 + 6 3 = (a + 6) 3 was written by him "a cubus + 

b in a quadr. 3 + a in b quadr. 3 + 6 cubo aequalia a + b cubo." 
In numerical equations the unknown quantity was denoted 
by N, its square by Q, and its cube by C. Thus the equation 
x 3 - 8x 2 + 16x = 40 was written \C - 8Q + 16 N (Equal. 40. 
Observe that exponents and our symbol (=) for equality 
were not yet in use; but that Vieta employed the Maltese 
cross (+) as the short-hand symbol for addition, and the (— ) 
for subtraction. These two characters had not been in general 
use before the time of Vieta. "It is very singular," saysHallam, 
"that discoveries of the greatest convenience, and, apparently, 
not above the ingenuity of a village schoolmaster, should 
have been overlooked by men of extraordinary acuteness like 
Tartaglia, Cardan, and Ferrari; and hardly less so that, by 
dint of that acuteness, they dispensed with the aid of these 
contrivances in which we suppose that so much of the utility 
of algebraic expression consists." Even after improvements 
in notation were once proposed, it was with extreme slowness 
that they were admitted into general use. They were made 
oftener by accident than design, and their authors had little 
notion of the effect of the change which they were making. 
The introduction of the + and — symbols seems to be due 
to the Germans, who, although they did not enrich algebra 
during the Renaissance with great inventions, as did the 
Italians, still cultivated it with great zeal. The arithmetic of 


John Widmann, printed A.D. 1489 in Leipzig, is the earliest 
book in which the + and — symbols have been found. There 
are indications leading us to surmise that they were in use 
first among merchants. They occur again in the arithmetic 
of Grammateus, a teacher at the University of Vienna. His 
pupil, Christoff Rudolff, the writer of the first text-book on 
algebra in the German language (printed in 1525), employs 
these symbols also. So did Stifel, who brought out a second 
edition of Rudolff' s Coss in 1553. Thus, by slow degrees, 
their adoption became universal. There is another short-hand 
symbol of which we owe the origin to the Germans. In a 
manuscript published sometime in the fifteenth century, a dot 
placed before a number is made to signify the extraction of a 
root of that number. This dot is the embryo of our present 
symbol for the square root. Christoff Rudolff, in his algebra, 
remarks that "the radix quadrata is, for brevity, designated in 
his algorithm with the character a/, as a/4." Here the dot has 
grown into a symbol much like our own. This same symbol 
was used by Michael Stifel. Our sign of equality is due to 
Robert Recorde (1510-1558), the author of The Whetstone 
of Witte (1557), which is the first English treatise on algebra. 
He selected this symbol because no two things could be more 
equal than two parallel lines =. The sign -=- for division was 
first used by Johann Heinrich Rahn, a Swiss, in 1659, and was 
introduced in England by John Pell in 1668. 

Michael Stifel (14867-1567), the greatest German alge- 
braist of the sixteenth century, was born in Esslingen, and 
died in Jena. He was educated in the monastery of his native 


place, and afterwards became Protestant minister. The study 
of the significance of mystic numbers in Revelation and in 
Daniel drew him to mathematics. He studied German and 
Italian works, and published in 1544, in Latin, a book entitled 
Arithmetica Integra. Melanchthon wrote a preface to it. Its 
three parts treat respectively of rational numbers, irrational 
numbers, and algebra. Stifel gives a table containing the 
numerical values of the binomial coefficients for powers below 
the 18th. He observes an advantage in letting a geometric 
progression correspond to an arithmetical progression, and 
arrives at the designation of integral powers by numbers. Here 
are the germs of the theory of exponents. In 1545 Stifel 
published an arithmetic in German. His edition of Rudolff 's 
Coss contains rules for solving cubic equations, derived from 
the writings of Cardan. 

We remarked above that Vieta discarded negative roots 
of equations. Indeed, we find few algebraists before and 
during the Renaissance who understood the significance even 
of negative quantities. Fibonacci seldom uses them. Pacioli 
states the rule that "minus times minus gives plus," but 
applies it really only to the development of the product 
of (a — b)(c — d); purely negative quantities do not appear 
in his work. The great German "Cossist" (algebraist), 
Michael Stifel, speaks as early as 1544 of numbers which 
are "absurd" or "fictitious below zero," and which arise 
when "real numbers above zero" are subtracted from zero. 
Cardan, at last, speaks of a "pure minus"; "but these ideas," 
says Hankel, "remained sparsely, and until the beginning of 


the seventeenth century, mathematicians dealt exclusively 
with absolute positive quantities." The first algebraist who 
occasionally places a purely negative quantity by itself on one 
side of an equation, is Harriot in England. As regards the 
recognition of negative roots, Cardan and Bombelli were far in 
advance of all writers of the Renaissance, including Vieta. Yet 
even they mentioned these so-called false or fictitious roots 
only in passing, and without grasping their real significance 
and importance. On this subject Cardan and Bombelli had 
advanced to about the same point as had the Hindoo Bhaskara, 
who saw negative roots, but did not approve of them. The 
generalisation of the conception of quantity so as to include 
the negative, was an exceedingly slow and difficult process in 
the development of algebra. 

We shall now consider the history of geometry during the 
Renaissance. Unlike algebra, it made hardly any progress. 
The greatest gain was a more intimate knowledge of Greek 
geometry. No essential progress was made before the time of 
Descartes. Regiomontanus, Xylander of Augsburg, Tartaglia, 
Commandinus of Urbino in Italy, Maurolycus, and others, 
made translations of geometrical works from the Greek. John 
Werner of Nurnberg published in 1522 the first work on conies 
which appeared in Christian Europe. Unlike the geometers 
of old, he studied the sections in relation with the cone, 
and derived their properties directly from it. This mode of 
studying the conies was followed by Maurolycus of Messina 
(1494-1575). The latter is, doubtless, the greatest geometer 
of the sixteenth century. From the notes of Pappus, he 


attempted to restore the missing fifth book of Apollonius on 
maxima and minima. His chief work is his masterly and 
original treatment of the conic sections, wherein he discusses 
tangents and asymptotes more fully than Apollonius had 
done, and applies them to various physical and astronomical 

The foremost geometrician of Portugal was Nonius; of 
France, before Vieta, was Peter Ramus, who perished in 
the massacre of St. Bartholomew. Vieta possessed great 
familiarity with ancient geometry. The new form which he 
gave to algebra, by representing general quantities by letters, 
enabled him to point out more easily how the construction 
of the roots of cubics depended upon the celebrated ancient 
problems of the duplication of the cube and the trisection 
of an angle. He reached the interesting conclusion that the 
former problem includes the solutions of all cubics in which 
the radical in Tartaglia's formula is real, but that the latter 
problem includes only those leading to the irreducible case. 

The problem of the quadrature of the circle was revived in 
this age, and was zealously studied even by men of eminence 
and mathematical ability. The army of circle-squarers became 
most formidable during the seventeenth century. Among 
the first to revive this problem was the German Cardinal 
Nicolaus Cusanus (died 1464), who had the reputation of 
being a great logician. His fallacies were exposed to full 
view by Regiomontanus. As in this case, so in others, every 
quadrator of note raised up an opposing mathematician: 
Orontius was met by Buteo and Nonius; Joseph Scaliger by 


Vieta, Adrianus Romanus, and Clavius; A. Quercu by Peter 
Metius. Two mathematicians of Netherlands, Adrianus 
Romanus and Ludolph van Ceulen, occupied themselves 
with approximating to the ratio between the circumference 
and the diameter. The former carried the value tt to 15, the 
latter to 35, places. The value of tv is therefore often named 
"Ludolph's number." His performance was considered so 
extraordinary, that the numbers were cut on his tomb-stone in 
St. Peter's church-yard, at Leyden. Romanus was the one who 
propounded for solution that equation of the forty-fifth degree 
solved by Vieta. On receiving Vieta's solution, he at once 
departed for Paris, to make his acquaintance with so great a 
master. Vieta proposed to him the Apollonian problem, to 
draw a circle touching three given circles. "Adrianus Romanus 
solved the problem by the intersection of two hyperbolas; but 
this solution did not possess the rigour of the ancient geometry. 
Vieta caused him to see this, and then, in his turn, presented a 
solution which had all the rigour desirable." [25] Romanus did 
much toward simplifying spherical trigonometry by reducing, 
by means of certain projections, the 28 cases in triangles then 
considered to only six. 

Mention must here be made of the improvements of the 
Julian calendar. The yearly determination of the movable 
feasts had for a long time been connected with an untold 
amount of confusion. The rapid progress of astronomy led 
to the consideration of this subject, and many new calendars 
were proposed. Pope Gregory XIII. convoked a large number 
of mathematicians, astronomers, and prelates, who decided 


upon the adoption of the calendar proposed by the Jesuit 
Lilius Clavius. To rectify the errors of the Julian calendar it 
was agreed to write in the new calendar the 15th of October 
immediately after the 4th of October of the year 1582. The 
Gregorian calendar met with a great deal of opposition both 
among scientists and among Protestants. Clavius, who ranked 
high as a geometer, met the objections of the former most 
ably and effectively; the prejudices of the latter passed away 
with time. 

The passion for the study of mystical properties of num- 
bers descended from the ancients to the moderns. Much 
was written on numerical mysticism even by such eminent 
men as Pacioli and Stifel. The Numerorum Mysteria of 
Peter Bungus covered 700 quarto pages. He worked with 
great industry and satisfaction on 666, which is the num- 
ber of the beast in Revelation (xiii. 18), the symbol of 
Antichrist. He reduced the name of the 'impious' Martin 
Luther to a form which may express this formidable number. 
Placing a = 1, 6 = 2, etc, k = 10, I — 20, etc., he finds, 
after misspelling the name, that M^A/^R^qjT^qoJ^N^o) 
L (20) V (200) T (i00) E (5) R (80) A (i) constitutes the number re- 
quired. These attacks on the great reformer were not 
unprovoked, for his friend, Michael Stifel, the most acute and 
original of the early mathematicians of Germany, exercised an 
equal ingenuity in showing that the above number referred to 
Pope Leo X., — a demonstration which gave Stifel unspeakable 
comfort. [22] 

Astrology also was still a favourite study. It is well known 


that Cardan, Maurolycus, Regiomontanus, and many other 
eminent scientists who lived at a period even later than this, 
engaged in deep astrological study; but it is not so generally 
known that besides the occult sciences already named, men 
engaged in the mystic study of star-polygons and magic 
squares. "The pentagramma gives you pain," says Faust to 
Mephistopheles. It is of deep psychological interest to see 
scientists, like the great Kepler, demonstrate on one page a 
theorem on star-polygons, with strict geometric rigour, while 
on the next page, perhaps, he explains their use as amulets 
or in conjurations. [1] Playfair, speaking of Cardan as an 
astrologer, calls him "a melancholy proof that there is no 
folly or weakness too great to be united to high intellectual 
attainments." [26] Let our judgment not be too harsh. The 
period under consideration is too near the Middle Ages to 
admit of complete emancipation from mysticism even among 
scientists. Scholars like Kepler, Napier, Albrecht Diirer, while 
in the van of progress and planting one foot upon the firm 
ground of truly scientific inquiry, were still resting with the 
other foot upon the scholastic ideas of preceding ages. 


The ecclesiastical power, which in the ignorant ages was 
an unmixed benefit, in more enlightened ages became a 
serious evil. Thus, in France, during the reigns preced- 
ing that of Henry IV., the theological spirit predominated. 
This is painfully shown by the massacres of Vassy and of 


St. Bartholomew. Being engaged in religious disputes, people 
had no leisure for science and for secular literature. Hence, 
down to the time of Henry IV., the French "had not put forth 
a single work, the destruction of which would now be a loss 
to Europe." In England, on the other hand, no religious 
wars were waged. The people were comparatively indifferent 
about religious strifes; they concentrated their ability upon 
secular matters, and acquired, in the sixteenth century, a 
literature which is immortalised by the genius of Shakespeare 
and Spenser. This great literary age in England was followed 
by a great scientific age. At the close of the sixteenth cen- 
tury, the shackles of ecclesiastical authority were thrown off 
by France. The ascension of Henry IV. to the throne was 
followed in 1598 by the Edict of Nantes, granting freedom of 
worship to the Huguenots, and thereby terminating religious 
wars. The genius of the French nation now began to blossom. 
Cardinal Richelieu, during the reign of Louis XIII., pursued 
the broad policy of not favouring the opinions of any sect, 
but of promoting the interests of the nation. His age was 
remarkable for the progress of knowledge. It produced that 
great secular literature, the counterpart of which was found 
in England in the sixteenth century. The seventeenth century 
was made illustrious also by the great French mathematicians, 
Roberval, Descartes, Desargues, Fermat, and Pascal. 

More gloomy is the picture in Germany. The great changes 
which revolutionised the world in the sixteenth century, and 
which led England to national greatness, led Germany to 
degradation. The first effects of the Reformation there 


were salutary. At the close of the fifteenth and during the 
sixteenth century, Germany had been conspicuous for her 
scientific pursuits. She had been the leader in astronomy 
and trigonometry. Algebra also, excepting for the discoveries 
in cubic equations, was, before the time of Vieta, in a more 
advanced state there than elsewhere. But at the beginning of 
the seventeenth century, when the sun of science began to rise 
in France, it set in Germany. Theologic disputes and religious 
strife ensued. The Thirty Years' War (1618-1648) proved 
ruinous. The German empire was shattered, and became 
a mere lax confederation of petty despotisms. Commerce 
was destroyed; national feeling died out. Art disappeared, 
and in literature there was only a slavish imitation of French 
artificiality. Nor did Germany recover from this low state 
for 200 years; for in 1756 began another struggle, the Seven 
Years' War, which turned Prussia into a wasted land. Thus 
it followed that at the beginning of the seventeenth century, 
the great Kepler was the only German mathematician of 
eminence, and that in the interval of 200 years between Kepler 
and Gauss, there arose no great mathematician in Germany 
excepting Leibniz. 

Up to the seventeenth century, mathematics was cultivated 
but little in Great Britain. During the sixteenth century, 
she brought forth no mathematician comparable with Vieta, 
Stifel, or Tartaglia. But with the time of Recorde, the English 
became conspicuous for numerical skill. The first important 
arithmetical work of English authorship was published in 
Latin in 1522 by Cuthbert Tonstall (1474-1559). He had 


studied at Oxford, Cambridge, and Padua, and drew freely 
from the works of Pacioli and Regiomontanus. Reprints 
of his arithmetic appeared in England and France. After 
Recorde the higher branches of mathematics began to be 
studied. Later, Scotland brought forth Napier, the inventor 
of logarithms. The instantaneous appreciation of their value 
is doubtless the result of superiority in calculation. In Italy, 
and especially in France, geometry, which for a long time had 
been an almost stationary science, began to be studied with 
success. Galileo, Torricelli, Roberval, Fermat, Desargues, 
Pascal, Descartes, and the English Wallis are the great 
revolutioners of this science. Theoretical mechanics began to 
be studied. The foundations were laid by Fermat and Pascal 
for the theory of numbers and the theory of probability. 

We shall first consider the improvements made in the 
art of calculating. The nations of antiquity experimented 
thousands of years upon numeral notations before they 
happened to strike upon the so-called "Arabic notation." In 
the simple expedient of the cipher, which was introduced by 
the Hindoos about the fifth or sixth century after Christ, 
mathematics received one of the most powerful impulses. 
It would seem that after the "Arabic notation" was once 
thoroughly understood, decimal fractions would occur at once 
as an obvious extension of it. But "it is curious to think how 
much science had attempted in physical research and how 
deeply numbers had been pondered, before it was perceived 
that the all-powerful simplicity of the 'Arabic notation' was 
as valuable and as manageable in an infinitely descending as 


in an infinitely ascending progression." [28] Simple as decimal 
fractions appear to us, the invention of them is not the result 
of one mind or even of one age. They came into use by almost 
imperceptible degrees. The first mathematicians identified 
with their history did not perceive their true nature and 
importance, and failed to invent a suitable notation. The idea 
of decimal fractions makes its first appearance in methods 
for approximating to the square roots of numbers. Thus 
John of Seville, presumably in imitation of Hindoo rules, adds 
2 n ciphers to the number, then finds the square root, and 
takes this as the numerator of a fraction whose denominator 
is 1 followed by n ciphers. The same method was followed 
by Cardan, but it failed to be generally adopted even by his 
Italian contemporaries; for otherwise it would certainly have 
been at least mentioned by Cataldi (died 1626) in a work 
devoted exclusively to the extraction of roots. Cataldi finds 
the square root by means of continued fractions — a method 
ingenious and novel, but for practical purposes inferior to 
Cardan's. Orontius Finaeus (died 1555) in France, and 
William Buckley (died about 1550) in England extracted 
the square root in the same way as Cardan and John of 
Seville. The invention of decimals is frequently attributed 
to Regiomontanus, on the ground that instead of placing the 
sinus totus, in trigonometry, equal to a multiple of 60, like the 
Greeks, he put it = 100, 000. But here the trigonometrical lines 
were expressed in integers, and not in fractions. Though he 
adopted a decimal division of the radius, he and his successors 
did not apply the idea outside of trigonometry and, indeed, 


had no notion whatever of decimal fractions. To Simon 

Stevin of Bruges in Belgium (1548-1620), a man who did 

a great deal of work in most diverse fields of science, we 

owe the first systematic treatment of decimal fractions. In 

his La Disme (1585) he describes in very express terms the 

advantages, not only of decimal fractions, but also of the 

decimal division in systems of weights and measures. Stevin 

applied the new fractions "to all the operations of ordinary 

arithmetic." [25] What he lacked was a suitable notation. 

In place of our decimal point, he used a cipher; to each 

place in the fraction was attached the corresponding index. 

Thus, in his notation, the number 5.912 would be 5912 or 

5® 9® 1©2(3). These indices, though cumbrous in practice, 

are of interest, because they are the germ of an important 

innovation. To Stevin belongs the honour of inventing our 

present mode of designating powers and also of introducing 

fractional exponents into algebra. Strictly speaking, this had 

been done much earlier by Oresme, but it remained wholly 

unnoticed. Not even Stevin's innovations were immediately 

appreciated or at once accepted, but, unlike Oresme's, they 

remained a secure possession. No improvement was made in 

the notation of decimals till the beginning of the seventeenth 

century. After Stevin, decimals were used by Joost Biirgi, 

a Swiss by birth, who prepared a manuscript on arithmetic 

soon after 1592, and by Johann Hartmann Beyer, who 

assumes the invention as his own. In 1603, he published at 

Frankfurt on the Main a Logistica Decimalis. With Biirgi, a 

zero placed underneath the digit in unit's place answers as 


sign of separation. Beyer's notation resembles Stevin's. The 
decimal point, says Peacock, is due to Napier, who in 1617 
published his Rabdologia, containing a treatise on decimals, 
wherein the decimal point is used in one or two instances. 
In the English translation of Napier's Mirifici logarithmorum 
canonis descriptio, executed by Edward Wright in 1616, and 
corrected by the author, the decimal point occurs in the 
tables. There is no mention of decimals in English arithmetics 
between 1619 and 1631. Oughtred in 1631 designates the 
fraction .56 thus, 1 56. Albert Girard, a pupil of Stevin, in 
1629 uses the point on one occasion. John Wallis in 1657 
writes 12 1 345 , but afterwards in his algebra adopts the usual 
point. De Morgan says that "to the first quarter of the 
eighteenth century we must refer not only the complete and 
final victory of the decimal point, but also that of the now 
universal method of performing the operations of division and 
extraction of the square root. [27] We have dwelt at some 
length on the progress of the decimal notation, because "the 
history of language ... is of the highest order of interest, as 
well as utility: its suggestions are the best lesson for the future 
which a reflecting mind can have." [27] 

The miraculous powers of modern calculation are due to 
three inventions: the Arabic Notation, Decimal Fractions, 
and Logarithms. The invention of logarithms in the first 
quarter of the seventeenth century was admirably timed, for 
Kepler was then examining planetary orbits, and Galileo had 
just turned the telescope to the stars. During the Renaissance 
German mathematicians had constructed trigonometrical 


tables of great accuracy, but this greater precision enormously 
increased the work of the calculator. It is no exaggeration 
to say that the invention of logarithms "by shortening the 
labours doubled the life of the astronomer." Logarithms were 
invented by John Napier, Baron of Merchiston, in Scotland 
(1550-1617). It is one of the greatest curiosities of the 
history of science that Napier constructed logarithms before 
exponents were used. To be sure, Stifel and Stevin made some 
attempts to denote powers by indices, but this notation was 
not generally known, — not even to Harriot, whose algebra 
appeared long after Napier's death. That logarithms flow 
naturally from the exponential symbol was not observed until 
much later. It was Euler who first considered logarithms as 
being indices of powers. What, then, was Napier's line of 

Let AB be a definite line, DE a line extending from 
D indefinitely. Imagine two points starting at the same 

moment; the one mov- 
A C B ing from A toward B, 

the other from D to- 

i 1 1 

D F E ward E. Let the ve- 

locity during the first 
moment be the same for both: let that of the point on line DE 
be uniform; but the velocity of the point on AB decreasing in 
such a way that when it arrives at any point C, its velocity is 
proportional to the remaining distance BC. While the first 
point moves over a distance AC, the second one moves over a 
distance DF. Napier calls DF the logarithm of BC. 


Napier's process is so unique and so different from all other 
modes of presenting the subject that there cannot be the 
shadow of a doubt that this invention is entirely his own; it 
is the result of unaided, isolated speculation. He first sought 
the logarithms only of sines; the line AB was the sine of 90° 
and was taken = 10 7 ; BC was the sine of the arc, and DF 
its logarithm. We notice that as the motion proceeds, BC 
decreases in geometrical progression, while DF increases in 
arithmetical progression. Let AB = a = 10 7 , let x = DF, 

y = BC, then AC = a — y. The velocity of the point C is 

d(a — y) . 

= y; this gives — nat. log y = t + c. When t = 0, then 

at , dx 

y = a and c = —nat. log a. Again, let — = a be the velocity 


of the point F, then x = at. Substituting for t and c their 
values and remembering that a = 10 7 and that by definition 
x = Nap. logy, we get 

7 10 7 
Nap. logy = 10 nat. log . 


It is evident from this formula that Napier's logarithms are 
not the same as the natural logarithms. Napier's logarithms 
increase as the number itself decreases. He took the logarithm 
of sin 90 = 0; i.e. the logarithm of 10 7 = 0. The logarithm of 
sin a increased from zero as a decreased from 90°. Napier's 
genesis of logarithms from the conception of two flowing points 
reminds us of Newton's doctrine of fluxions. The relation 
between geometric and arithmetical progressions, so skilfully 
utilised by Napier, had been observed by Archimedes, Stifel, 
and others. Napier did not determine the base to his system 
of logarithms. The notion of a "base" in fact never suggested 


itself to him. The one demanded by his reasoning is the 
reciprocal of that of the natural system, but such a base would 
not reproduce accurately all of Napier's figures, owing to slight 
inaccuracies in the calculation of the tables. Napier's great 
invention was given to the world in 1614 in a work entitled 
Mirifici logarithmorum canonis descriptio. In it he explained 
the nature of his logarithms, and gave a logarithmic table of 
the natural sines of a quadrant from minute to minute. 

Henry Briggs (1556-1631), in Napier's time professor 
of geometry at Gresham College, London, and afterwards 
professor at Oxford, was so struck with admiration of Napier's 
book, that he left his studies in London to do homage to 
the Scottish philosopher. Briggs was delayed in his journey, 
and Napier complained to a common friend, "Ah, John, 
Mr. Briggs will not come." At that very moment knocks 
were heard at the gate, and Briggs was brought into the 
lord's chamber. Almost one-quarter of an hour was spent, 
each beholding the other without speaking a word. At 
last Briggs began: "My lord, I have undertaken this long 
journey purposely to see your person, and to know by what 
engine of wit or ingenuity you came first to think of this 
most excellent help in astronomy, viz. the logarithms; but, 
my lord, being by you found out, I wonder nobody found 
it out before, when now known it is so easy." [28] Briggs 
suggested to Napier the advantage that would result from 
retaining zero for the logarithm of the whole sine, but choosing 
10, 000, 000, 000 for the logarithm of the 10th part of that same 
sine, i. e. of 5° 44 / 22 // . Napier said that he had already thought 


of the change, and he pointed out a slight improvement on 
Briggs' idea; viz. that zero should be the logarithm of 1, and 
10, 000, 000, 000 that of the whole sine, thereby making the 
characteristic of numbers greater than unity positive and not 
negative, as suggested by Briggs. Briggs admitted this to 
be more convenient. The invention of "Briggian logarithms" 
occurred, therefore, to Briggs and Napier independently. The 
great practical advantage of the new system was that its 
fundamental progression was accommodated to the base, 10, 
of our numerical scale. Briggs devoted all his energies to the 
construction of tables upon the new plan. Napier died in 
1617, with the satisfaction of having found in Briggs an able 
friend to bring to completion his unfinished plans. In 1624 
Briggs published his Arithmetica logarithmica, containing the 
logarithms to 14 places of numbers, from 1 to 20, 000 and from 
90, 000 to 100, 000. The gap from 20, 000 to 90, 000 was filled 
up by that illustrious successor of Napier and Briggs, Adrian 
Vlacq of Gouda in Holland. He published in 1628 a table of 
logarithms from 1 to 100, 000, of which 70, 000 were calculated 
by himself. The first publication of Briggian logarithms 
of trigonometric functions was made in 1620 by Gunter, 
a colleague of Briggs, who found the logarithmic sines and 
tangents for every minute to seven places. Gunter was the 
inventor of the words cosine and cotangent. Briggs devoted 
the last years of his life to calculating more extensive Briggian 
logarithms of trigonometric functions, but he died in 1631, 
leaving his work unfinished. It was carried on by the English 
Henry Gellibrand, and then published by Vlacq at his own 


expense. Briggs divided a degree into 100 parts, but owing to 
the publication by Vlacq of trigonometrical tables constructed 
on the old sexagesimal division, Briggs' innovation remained 
unrecognised. Briggs and Vlacq published four fundamental 
works, the results of which "have never been superseded by 
any subsequent calculations." 

The first logarithms upon the natural base e were published 
by John Speidell in his New Logarithmes (London, 1619), 
which contains the natural logarithms of sines, tangents, and 

The only possible rival of John Napier in the invention 
of logarithms was the Swiss Justus Byrgius (Joost Biirgi). 
He published a rude table of logarithms six years after the 
appearance of the Canon Mirificus, but it appears that he 
conceived the idea and constructed that table as early, if not 
earlier, than Napier did his. But he neglected to have the 
results published until Napier's logarithms were known and 
admired throughout Europe. 

Among the various inventions of Napier to assist the memory 
of the student or calculator, is "Napier's rule of circular parts" 
for the solution of spherical right triangles. It is, perhaps, 
"the happiest example of artificial memory that is known." 

The most brilliant conquest in algebra during the sixteenth 
century had been the solution of cubic and biquadratic 
equations. All attempts at solving algebraically equations of 
higher degrees remaining fruitless, a new line of inquiry — the 
properties of equations and their roots — was gradually opened 
up. We have seen that Vieta had attained a partial knowledge 


of the relations between roots and coefficients. Peletarius, a 
Frenchman, had observed as early as 1558, that the root of an 
equation is a divisor of the last term. One who extended the 
theory of equations somewhat further than Vieta, was Albert 
Girard (1590-1634), a Flemish mathematician. Like Vieta, 
this ingenious author applied algebra to geometry, and was the 
first who understood the use of negative roots in the solution 
of geometric problems. He spoke of imaginary quantities; 
inferred by induction that every equation has as many roots as 
there are units in the number expressing its degree; and first 
showed how to express the sums of their powers in terms of the 
coefficients. Another algebraist of considerable power was the 
English Thomas Harriot (1560-1621). He accompanied the 
first colony sent out by Sir Walter Raleigh to Virginia. After 
having surveyed that country he returned to England. As a 
mathematician, he was the boast of his country. He brought 
the theory of equations under one comprehensive point of 
view by grasping that truth in its full extent to which Vieta 
and Girard only approximated; viz. that in an equation in its 
simplest form, the coefficient of the second term with its sign 
changed is equal to the sum of the roots; the coefficient of the 
third is equal to the sum of the products of every two of the 
roots; etc. He was the first to decompose equations into their 
simple factors; but, since he failed to recognise imaginary and 
even negative roots, he failed also to prove that every equation 
could be thus decomposed. Harriot made some changes in 
algebraic notation, adopting small letters of the alphabet in 
place of the capitals used by Vieta. The symbols of inequality 


> and < were introduced by him. Harriot's work, Artis 
Analytics praxis, was published in 1631, ten years after his 
death. William Oughtred (1574-1660) contributed vastly 
to the propagation of mathematical knowledge in England by 
his treatises, which were long used in the universities. He 
introduced x as symbol of multiplication, and :: as that of 
proportion. By him ratio was expressed by only one dot. In 
the eighteenth century Christian Wolf secured the general 
adoption of the dot as a symbol of multiplication, and the 
sign for ratio was thereupon changed to two dots. Oughtred's 
ministerial duties left him but little time for the pursuit of 
mathematics during daytime, and evenings his economical 
wife denied him the use of a light. 

Algebra was now in a state of sufficient perfection to enable 
Descartes to take that important step which forms one of the 
grand epochs in the history of mathematics, — the application 
of algebraic analysis to define the nature and investigate the 
properties of algebraic curves. 

In geometry, the determination of the areas of curvilinear 
figures was diligently studied at this period. Paul Guldin 
(1577-1643), a Swiss mathematician of considerable note, 
re-discovered the following theorem, published in his Centro- 
baryca, which has been named after him, though first found 
in the Mathematical Collections of Pappus: The volume of a 
solid of revolution is equal to the area of the generating figure, 
multiplied by the circumference described by the centre of 
gravity. We shall see that this method excels that of Kepler 
and Cavalieri in following a more exact and natural course; 


but it has the disadvantage of necessitating the determination 
of the centre of gravity, which in itself may be a more difficult 
problem than the original one of finding the volume. Guldin 
made some attempts to prove his theorem, but Cavalieri 
pointed out the weakness of his demonstration. 

Johannes Kepler (1571-1630) was a native of Wiirtem- 
berg and imbibed Copernican principles while at the Uni- 
versity of Tubingen. His pursuit of science was repeatedly 
interrupted by war, religious persecution, pecuniary embar- 
rassments, frequent changes of residence, and family troubles. 
In 1600 he became for one year assistant to the Danish as- 
tronomer, Tycho Brahe, in the observatory near Prague. The 
relation between the two great astronomers was not always of 
an agreeable character. Kepler's publications are voluminous. 
His first attempt to explain the solar system was made in 1596, 
when he thought he had discovered a curious relation between 
the five regular solids and the number and distance of the 
planets. The publication of this pseudo- discovery brought 
him much fame. Maturer reflection and intercourse with 
Tycho Brahe and Galileo led him to investigations and results 
more worthy of his genius — "Kepler's laws." He enriched 
pure mathematics as well as astronomy. It is not strange 
that he was interested in the mathematical science which 
had done him so much service; for "if the Greeks had not 
cultivated conic sections, Kepler could not have superseded 
Ptolemy." [11] The Greeks never dreamed that these curves 
would ever be of practical use; Aristaeus and Apollonius 
studied them merely to satisfy their intellectual cravings after 


the ideal; yet the conic sections assisted Kepler in tracing the 
march of the planets in their elliptic orbits. Kepler made also 
extended use of logarithms and decimal fractions, and was 
enthusiastic in diffusing a knowledge of them. At one time, 
while purchasing wine, he was struck by the inaccuracy of 
the ordinary modes of determining the contents of kegs. This 
led him to the study of the volumes of solids of revolution 
and to the publication of the Stereometria Doliorum in 1615. 
In it he deals first with the solids known to Archimedes 
and then takes up others. Kepler introduced a new idea 
into geometry; namely, that of infinitely great and infinitely 
small quantities. Greek mathematicians always shunned this 
notion, but with it modern mathematicians have completely 
revolutionised the science. In comparing rectilinear figures, 
the method of superposition was employed by the ancients, 
but in comparing rectilinear and curvilinear figures with each 
other, this method failed because no addition or subtraction 
of rectilinear figures could ever produce curvilinear ones. To 
meet this case, they devised the Method of Exhaustion, which 
was long and difficult; it was purely synthetical, and in general 
required that the conclusion should be known at the outset. 
The new notion of infinity led gradually to the invention of 
methods immeasurably more powerful. Kepler conceived the 
circle to be composed of an infinite number of triangles having 
their common vertices at the centre, and their bases in the 
circumference; and the sphere to consist of an infinite number 
of pyramids. He applied conceptions of this kind to the 
determination of the areas and volumes of figures generated 


by curves revolving about any line as axis, but succeeded in 
solving only a few of the simplest out of the 84 problems which 
he proposed for investigation in his Stereometria. 

Other points of mathematical interest in Kepler's works are 

(1) the statement of the earliest problem of inverse tangents; 

(2) an investigation which amounts to the evaluation of the 

definite integral / sin <fi d<j> = 1 — cos <fi; (3) the assertion that 

the circumference of an ellipse, whose axes are 2a and 26, 

is nearly n(a + b); (4) a passage from which it has been 

inferred that Kepler knew the variation of a function near 

its maximum value to disappear; (5) the assumption of the 

principle of continuity (which differentiates modern from 

ancient geometry), when he shows that a parabola has a focus 

at infinity, that lines radiating from this "csecus focus" are 

parallel and have no other point at infinity 

The Stereometria led Cavalieri, an Italian Jesuit, to the 

consideration of infinitely small quantities. Bonaventura 

Cavalieri (1598-1647), a pupil of Galileo and professor 

at Bologna, is celebrated for his Geometria indivisibilibus 

continuorum nova quadam ratione promota, 1635. This 

work expounds his method of Indivisibles, which occupies an 

intermediate place between the method of exhaustion of the 

Greeks and the methods of Newton and Leibniz. He considers 

lines as composed of an infinite number of points, surfaces 

as composed of an infinite number of lines, and solids of an 

infinite number of planes. The relative magnitude of two solids 

or surfaces could then be found simply by the summation of 


series of planes or lines. For example, he finds the sum of the 
squares of all lines making up a triangle equal to one-third the 
sum of the squares of all lines of a parallelogram of equal base 
and altitude; for if in a triangle, the first line at the apex be 1, 
then the second is 2, the third is 3, and so on; and the sum of 
their squares is 

l 2 + 2 2 + 3 2 + • • • + n 2 = n(n + l)(2n + 1) -=- 6. 

In the parallelogram, each of the lines is n and their number 
is n; hence the total sum of their squares is n 3 . The ratio 
between the two sums is therefore 

n(n+ l)(2n+ 1) 4- 6n 3 = \, 

since n is infinite. From this he concludes that the pyramid or 
cone is respectively \ of a prism or cylinder of equal base and 
altitude, since the polygons or circles composing the former 
decrease from the base to the apex in the same way as the 
squares of the lines parallel to the base in a triangle decrease 
from base to apex. By the Method of Indivisibles, Cavalieri 
solved the majority of the problems proposed by Kepler. 
Though expeditious and yielding correct results, Cavalieri's 
method lacks a scientific foundation. If a line has absolutely no 
width, then no number, however great, of lines can ever make 
up an area; if a plane has no thickness whatever, then even 
an infinite number of planes cannot form a solid. The reason 
why this method led to correct conclusions is that one area is 
to another area in the same ratio as the sum of the series of 
lines in the one is to the sum of the series of lines in the other. 


Though unscientific, Cavalieri's method was used for fifty 
years as a sort of integral calculus. It yielded solutions to some 
difficult problems. Guldin made a severe attack on Cavalieri 
and his method. The latter published in 1647, after the death 
of Guldin, a treatise entitled Exercitationes geometries sex, 
in which he replied to the objections of his opponent and 
attempted to give a clearer explanation of his method. Guldin 
had never been able to demonstrate the theorem named after 
him, except by metaphysical reasoning, but Cavalieri proved 
it by the method of indivisibles. A revised edition of the 
Geometry of Indivisibles appeared in 1653. 

There is an important curve, not known to the ancients, 
which now began to be studied with great zeal. Roberval 
gave it the name of "trochoid," Pascal the name of "roulette," 
Galileo the name of "cycloid." The invention of this curve 
seems to be due to Galileo, who valued it for the graceful 
form it would give to arches in architecture. He ascertained 
its area by weighing paper figures of the cycloid against that 
of the generating circle, and found thereby the first area to 
be nearly but not exactly thrice the latter. A mathematical 
determination was made by his pupil, Evangelista Torricelli 
(1608-1647), who is more widely known as a physicist than as 
a mathematician. 

By the Method of Indivisibles he demonstrated its area 
to be triple that of the revolving circle, and published his 
solution. This same quadrature had been effected a few years 
earlier by Roberval in France, but his solution was not known 
to the Italians. Roberval, being a man of irritable and violent 


disposition, unjustly accused the mild and amiable Torricelli 
of stealing the proof. This accusation of plagiarism created so 
much chagrin with Torricelli that it is considered to have been 
the cause of his early death. Vincenzo Viviani, another 
prominent pupil of Galileo, determined the tangent to the 
cycloid. This was accomplished in France by Descartes and 

In France, where geometry began to be cultivated with 
greatest success, Roberval, Fermat, Pascal, employed the 
Method of Indivisibles and made new improvements in it. 
Giles Persone de Roberval (1602-1675), for forty years 
professor of mathematics at the College of France in Paris, 
claimed for himself the invention of the Method of Indivisibles. 
Since his complete works were not published until after his 
death, it is difficult to settle questions of priority. Montucla 
and Chasles are of the opinion that he invented the method 
independent of and earlier than the Italian geometer, though 
the work of the latter was published much earlier than 
Roberval's. Marie finds it difficult to believe that the 
Frenchman borrowed nothing whatever from the Italian, 
for both could not have hit independently upon the word 
Indivisibles, which is applicable to infinitely small quantities, 
as conceived by Cavalieri, but not as conceived by Roberval. 
Roberval and Pascal improved the rational basis of the 
Method of Indivisibles, by considering an area as made up 
of an indefinite number of rectangles instead of lines, and 
a solid as composed of indefinitely small solids instead of 
surfaces. Roberval applied the method to the finding of areas, 


volumes, and centres of gravity. He effected the quadrature of 
a parabola of any degree y m = a m ~ 1 x, and also of a parabola 
yvn _ a m-n x n_ yy e ] iave a i reac iy mentioned his quadrature 

of the cycloid. Roberval is best known for his method of 
drawing tangents. He was the first to apply motion to the 
resolution of this important problem. His method is allied 
to Newton's principle of fluxions. Archimedes conceived his 
spiral to be generated by a double motion. This idea Roberval 
extended to all curves. Plane curves, as for instance the conic 
sections, may be generated by a point acted upon by two 
forces, and are the resultant of two motions. If at any point of 
the curve the resultant be resolved into its components, then 
the diagonal of the parallelogram determined by them is the 
tangent to the curve at that point. The greatest difficulty 
connected with this ingenious method consisted in resolving 
the resultant into components having the proper lengths and 
directions. Roberval did not always succeed in doing this, yet 
his new idea was a great step in advance. He broke off from 
the ancient definition of a tangent as a straight line having 
only one point in common with a curve, — a definition not 
valid for curves of higher degrees, nor apt even in curves of 
the second degree to bring out the properties of tangents and 
the parts they may be made to play in the generation of the 
curves. The subject of tangents received special attention 
also from Fermat, Descartes, and Barrow, and reached its 
highest development after the invention of the differential 
calculus. Fermat and Descartes defined tangents as secants 
whose two points of intersection with the curve coincide; 


Barrow considered a curve a polygon, and called one of its 
sides produced a tangent. 

A profound scholar in all branches of learning and a 
mathematician of exceptional powers was Pierre de Fermat 
(1601-1665). He studied law at Toulouse, and in 1631 was 
made councillor for the parliament of Toulouse. His leisure 
time was mostly devoted to mathematics, which he studied 
with irresistible passion. Unlike Descartes and Pascal, he led a 
quiet and unaggressive life. Fermat has left the impress of his 
genius upon all branches of mathematics then known. A great 
contribution to geometry was his De maximis et minimis. 
About twenty years earlier, Kepler had first observed that 
the increment of a variable, as, for instance, the ordinate of 
a curve, is evanescent for values very near a maximum or a 
minimum value of the variable. Developing this idea, Fermat 
obtained his rule for maxima and minima. He substituted 
x + e for x in the given function of x and then equated to each 
other the two consecutive values of the function and divided 
the equation by e. If e be taken 0, then the roots of this 
equation are the values of x, making the function a maximum 
or a minimum. Fermat was in possession of this rule in 1629. 
The main difference between it and the rule of the differential 
calculus is that it introduces the indefinite quantity e instead 
of the infinitely small dx. Fermat made it the basis for his 
method of drawing tangents. 

Owing to a want of explicitness in statement, Fermat's 
method of maxima and minima, and of tangents, was severely 
attacked by his great contemporary, Descartes, who could 


never be brought to render due justice to his merit. In 
the ensuing dispute, Fermat found two zealous defenders in 
Roberval and Pascal, the father; while Mydorge, Desargues, 
and Hardy supported Descartes. 

Since Fermat introduced the conception of infinitely small 
differences between consecutive values of a function and 
arrived at the principle for finding the maxima and minima, 
it was maintained by Lagrange, Laplace, and Fourier, that 
Fermat may be regarded as the first inventor of the differential 
calculus. This point is not well taken, as will be seen from the 
words of Poisson, himself a Frenchman, who rightly says that 
the differential calculus "consists in a system of rules proper 
for finding the differentials of all functions, rather than in the 
use which may be made of these infinitely small variations in 
the solution of one or two isolated problems." 

A contemporary mathematician, whose genius excelled even 
that of the great Fermat, was Blaise Pascal (1623-1662). 
He was born at Clermont in Auvergne. In 1626 his father 
retired to Paris, where he devoted himself to teaching his 
son, for he would not trust his education to others. Blaise 
Pascal's genius for geometry showed itself when he was but 
twelve years old. His father was well skilled in mathematics, 
but did not wish his son to study it until he was perfectly 
acquainted with Latin and Greek. All mathematical books 
were hidden out of his sight. The boy once asked his father 
what mathematics treated of, and was answered, in general, 
"that it was the method of making figures with exactness, 
and of finding out what proportions they relatively had to one 


another." He was at the same time forbidden to talk any more 
about it, or ever to think of it. But his genius could not submit 
to be confined within these bounds. Starting with the bare 
fact that mathematics taught the means of making figures 
infallibly exact, he employed his thoughts about it and with a 
piece of charcoal drew figures upon the tiles of the pavement, 
trying the methods of drawing, for example, an exact circle or 
equilateral triangle. He gave names of his own to these figures 
and then formed axioms, and, in short, came to make perfect 
demonstrations. In this way he arrived unaided at the theorem 
that the sum of the three angles of a triangle is equal to two 
right angles. His father caught him in the act of studying this 
theorem, and was so astonished at the sublimity and force 
of his genius as to weep for joy. The father now gave him 
Euclid's Elements, which he, without assistance, mastered 
easily. His regular studies being languages, the boy employed 
only his hours of amusement on the study of geometry, yet 
he had so ready and lively a penetration that, at the age of 
sixteen, he wrote a treatise upon conies, which passed for such 
a surprising effort of genius, that it was said nothing equal to it 
in strength had been produced since the time of Archimedes. 
Descartes refused to believe that it was written by one so 
young as Pascal. This treatise was never published, and is 
now lost. Leibniz saw it in Paris and reported on a portion 
of its contents. The precocious youth made vast progress in 
all the sciences, but the constant application at so tender an 
age greatly impaired his health. Yet he continued working, 
and at nineteen invented his famous machine for performing 


arithmetical operations mechanically. This continued strain 
from overwork resulted in a permanent indisposition, and he 
would sometimes say that from the time he was eighteen, he 
never passed a day free from pain. At the age of twenty- four 
he resolved to lay aside the study of the human sciences and 
to consecrate his talents to religion. His Provincial Letters 
against the Jesuits are celebrated. But at times he returned 
to the favourite study of his youth. Being kept awake one 
night by a toothache, some thoughts undesignedly came into 
his head concerning the roulette or cycloid; one idea followed 
another; and he thus discovered properties of this curve 
even to demonstration. A correspondence between him and 
Fermat on certain problems was the beginning of the theory 
of probability. Pascal's illness increased, and he died at Paris 
at the early age of thirty- nine years. [30] By him the answer 
to the objection to Cavalieri's Method of Indivisibles was put 
in the clearest form. Like Roberval, he explained "the sum of 
right lines" to mean "the sum of infinitely small rectangles." 
Pascal greatly advanced the knowledge of the cycloid. He 
determined the area of a section produced by any line parallel 
to the base; the volume generated by it revolving around its 
base or around the axis; and, finally, the centres of gravity of 
these volumes, and also of half these volumes cut by planes 
of symmetry. Before publishing his results, he sent, in 1658, 
to all mathematicians that famous challenge offering prizes 
for the first two solutions of these problems. Only Wallis and 
A. La Louere competed for them. The latter was quite unequal 
to the task; the former, being pressed for time, made numerous 


mistakes: neither got a prize. Pascal then published his own 
solutions, which produced a great sensation among scientific 
men. Wallis, too, published his, with the errors corrected. 
Though not competing for the prizes, Huygens, Wren, and 
Fermat solved some of the questions. The chief discoveries 
of Christopher Wren (1632-1723), the celebrated architect 
of St. Paul's Cathedral in London, were the rectification of a 
cycloidal arc and the determination of its centre of gravity. 
Fermat found the area generated by an arc of the cycloid. 
Huygens invented the cycloidal pendulum. 

The beginning of the seventeenth century witnessed also a 
revival of synthetic geometry. One who treated conies still 
by ancient methods, but who succeeded in greatly simplifying 
many prolix proofs of Apollonius, was Claude Mydorge in 
Paris (1585-1647), a friend of Descartes. But it remained for 
Girard Desargues (1593-1662) of Lyons, and for Pascal, 
to leave the beaten track and cut out fresh paths. They 
introduced the important method of Perspective. All conies 
on a cone with circular base appear circular to an eye at the 
apex. Hence Desargues and Pascal conceived the treatment of 
the conic sections as projections of circles. Two important and 
beautiful theorems were given by Desargues: The one is on 
the "involution of the six points," in which a transversal meets 
a conic and an inscribed quadrangle; the other is that, if the 
vertices of two triangles, situated either in space or in a plane, 
lie on three lines meeting in a point, then their sides meet in 
three points lying on a line; and conversely. This last theorem 
has been employed in recent times by Brianchon, Sturm, 


Gergonne, and Poncelet. Poncelet made it the basis of his 
beautiful theory of homoligical figures. We owe to Desargues 
the theory of involution and of transversals; also the beautiful 
conception that the two extremities of a straight line may be 
considered as meeting at infinity, and that parallels differ from 
other pairs of lines only in having their points of intersection 
at infinity. Pascal greatly admired Desargues' results, saying 
(in his Essais pour les Coniques), "I wish to acknowledge that 
I owe the little that I have discovered on this subject, to his 
writings." Pascal's and Desargues' writings contained the 
fundamental ideas of modern synthetic geometry. In Pascal's 
wonderful work on conies, written at the age of sixteen and 
now lost, were given the theorem on the anharmonic ratio, 
first found in Pappus, and also that celebrated proposition on 
the mystic hexagon, known as "Pascal's theorem," viz. that 
the opposite sides of a hexagon inscribed in a conic intersect 
in three points which are collinear. This theorem formed 
the keystone to his theory. He himself said that from this 
alone he deduced over 400 corollaries, embracing the conies 
of Apollonius and many other results. Thus the genius of 
Desargues and Pascal uncovered several of the rich treasures 
of modern synthetic geometry; but owing to the absorbing 
interest taken in the analytical geometry of Descartes and later 
in the differential calculus, the subject was almost entirely 
neglected until the present century. 

In the theory of numbers no new results of scientific value 
had been reached for over 1000 years, extending from the 
times of Diophantus and the Hindoos until the beginning of 


the seventeenth century. But the illustrious period we are 
now considering produced men who rescued this science from 
the realm of mysticism and superstition, in which it had been 
so long imprisoned; the properties of numbers began again 
to be studied scientifically. Not being in possession of the 
Hindoo indeterminate analysis, many beautiful results of the 
Brahmins had to be re-discovered by the Europeans. Thus 
a solution in integers of linear indeterminate equations was 
re-discovered by the Frenchman Bachet de Meziriac (1581- 
1638), who was the earliest noteworthy European Diophantist. 
In 1612 he published Problemes plaisants et delectables qui se 
font par lesnombres, and in 1621 a Greek edition of Diophantus 
with notes. The father of the modern theory of numbers is 
Fermat. He was so uncommunicative in disposition, that he 
generally concealed his methods and made known his results 
only. In some cases later analysts have been greatly puzzled in 
the attempt of supplying the proofs. Fermat owned a copy of 
Bachet's Diophantus, in which he entered numerous marginal 
notes. In 1670 these notes were incorporated in a new edition 
of Diophantus, brought out by his son. Other theorems on 
numbers, due to Fermat, were published in his Opera varia 
(edited by his son) and in Wallis's Commercium epistolicum 
of 1658. Of the following theorems, the first seven are found 
in the marginal notes:— 

(1) x n + y n = z n is impossible for integral values of x, y, 
and z, when n > 2. Remark: "I have found for this a 
truly wonderful proof, but the margin is too small to hold 
it." Repeatedly was this theorem made the prize question 


of learned societies. It has given rise to investigations of 
great interest and difficulty on the part of Euler, Lagrange, 
Dirichlet, and Kummer. 

(2) A prime of the form An + 1 is only once the hypothenuse 
of a right triangle; its square is twice; its cube is three times, 
etc. Example: 5 2 = 3 2 + 4 2 ; 25 2 = 15 2 + 20 2 = 7 2 + 24 2 ; 
125 2 = 75 2 + 100 2 = 35 2 + 120 2 = 44 2 + 117 2 . 

(3) A prime of the form An + 1 can be expressed once, and 
only once, as the sum of two squares. Proved by Euler. 

(4) A number composed of two cubes can be resolved into 
two other cubes in an infinite multiplicity of ways. 

(5) Every number is either a triangular number or the sum 
of two or three triangular numbers; either a square or the sum 
of two, three, or four squares; either a pentagonal number 
or the sum of two, three, four, or five pentagonal numbers; 
similarly for polygonal numbers in general. The proof of 
this and other theorems is promised by Fermat in a future 
work which never appeared. This theorem is also given, with 
others, in a letter of 1637(?) addressed to Pater Mersenne. 

(6) As many numbers as you please may be found, such 
that the square of each remains a square on the addition to or 
subtraction from it of the sum of all the numbers. 

(7) x 4 + y 4 = z 1 is impossible. 

(8) In a letter of 1640 he gives the celebrated theorem 
generally known as "Fermat's theorem," which we state in 
Gauss's notation: If p is prime, and a is prime to p, then 
a p_1 = 1 (mod p). It was proved by Euler. 


(9) Fermat died with the belief that he had found a long- 
sought-for law of prime numbers in the formula 2 2 + 1 = 
a prime, but he admitted that he was unable to prove it 
rigorously. The law is not true, as was pointed out by Euler 
in the example 2 25 + 1 = 4, 294, 967, 297 = 6, 700, 417 times 641. 
The American lightning calculator Zerah Colburn, when a 
boy, readily found the factors, but was unable to explain the 
method by which he made his marvellous mental computation. 

(10) An odd prime number can be expressed as the difference 
of two squares in one, and only one, way. This theorem, given 
in the Relation, was used by Fermat for the decomposition of 
large numbers into prime factors. 

(11) If the integers a, 6, c represent the sides of a right 
triangle, then its area cannot be a square number. This was 
proved by Lagrange. 

(12) Fermat's solution of ax 2 + 1 = y 2 , where a is integral 
but not a square, has come down in only the broadest outline, 
as given in the Relation. He proposed the problem to the 
Frenchman, Bernhard Frenicle de Bessy, and in 1657 to all 
living mathematicians. In England, Wallis and Lord Brounker 
conjointly found a laborious solution, which was published 
in 1658, and also in 1668, in an algebraical work brought out 
by John Pell. Though Pell had no other connection with the 
problem, it went by the name of "Pell's problem." The first 
solution was given by the Hindoos. 

We are not sure that Fermat subjected all his theorems 
to rigorous proof. His methods of proof were entirely lost 
until 1879, when a document was found buried among the 


manuscripts of Huygens in the library of Leyden, entitled 
Relation des decouvertes en la science des nombres. It appears 
from it that he used an inductive method, called by him la 
descente infinie ou indefinie. He says that this was particularly 
applicable in proving the impossibility of certain relations, as, 
for instance, Theorem 11, given above, but that he succeeded 
in using the method also in proving affirmative statements. 
Thus he proved Theorem 3 by showing that if we suppose 
there be a prime 4n + 1 which does not possess this property, 
then there will be a smaller prime of the form 4n + 1 not 
possessing it; and a third one smaller than the second, not 
possessing it; and so on. Thus descending indefinitely, he 
arrives at the number 5, which is the smallest prime factor of 
the form 4n + 1. From the above supposition it would follow 
that 5 is not the sum of two squares — a conclusion contrary 
to fact. Hence the supposition is false, and the theorem 
is established. Fermat applied this method of descent with 
success in a large number of theorems. By this method Euler, 
Legendre, Dirichlet, proved several of his enunciations and 
many other numerical propositions. 

A correspondence between Pascal and Fermat relating to 
a certain game of chance was the germ of the theory of prob- 
abilities, which has since attained a vast growth. Chevalier 
de Mere proposed to Pascal the fundamental problem, to 
determine the probability which each player has, at any given 
stage of the game, of winning the game. Pascal and Fermat 
supposed that the players have equal chances of winning a 
single point. 


The former communicated this problem to Fermat, who 
studied it with lively interest and solved it by the theory 
of combinations, a theory which was diligently studied both 
by him and Pascal. The calculus of probabilities engaged 
the attention also of Huygens. The most important theorem 
reached by him was that, if A has p chances of winning a 

sum a, and q chances of winning a sum b, then he may expect 

dT) ~\~ bo 

to win the sum . The next great work on the theory of 

p + q 
probability was the Ars conjectandi of Jakob Bernoulli. 

Among the ancients, Archimedes was the only one who 
attained clear and correct notions on theoretical statics. He 
had acquired firm possession of the idea of pressure, which 
lies at the root of mechanical science. But his ideas slept 
nearly twenty centuries, until the time of Stevin and Galileo. 
Stevin determined accurately the force necessary to sustain 
a body on a plane inclined at any angle to the horizon. 
He was in possession of a complete doctrine of equilibrium. 
While Stevin investigated statics, Galileo pursued principally 
dynamics. Galileo was the first to abandon the Aristotelian 
idea that bodies descend more quickly in proportion as they 
are heavier; he established the first law of motion; determined 
the laws of falling bodies; and, having obtained a clear notion 
of acceleration and of the independence of different motions, 
was able to prove that projectiles move in parabolic curves. Up 
to his time it was believed that a cannon-ball moved forward 
at first in a straight line and then suddenly fell vertically 
to the ground. Galileo had an understanding of centrifugal 
forces, and gave a correct definition of momentum. Though 


he formulated the fundamental principle of statics, known as 
the parallelogram of forces, yet he did not fully recognise its 
scope. The principle of virtual velocities was partly conceived 
by Guido Ubaldo (died 1607), and afterwards more fully by 

Galileo is the founder of the science of dynamics. Among 
his contemporaries it was chiefly the novelties he detected 
in the sky that made him celebrated, but Lagrange claims 
that his astronomical discoveries required only a telescope 
and perseverance, while it took an extraordinary genius to 
discover laws from phenomena, which we see constantly and 
of which the true explanation escaped all earlier philosophers. 
The first contributor to the science of mechanics after Galileo 
was Descartes. 


Among the earliest thinkers of the seventeenth and eigh- 
teenth centuries, who employed their mental powers toward 
the destruction of old ideas and the up-building of new ones, 
ranks Rene Descartes (1596-1650). Though he professed 
orthodoxy in faith all his life, yet in science he was a profound 
sceptic. He found that the world's brightest thinkers had been 
long exercised in metaphysics, yet they had discovered nothing 
certain; nay, had even flatly contradicted each other. This led 
him to the gigantic resolution of taking nothing whatever on 
authority, but of subjecting everything to scrutinous exami- 
nation, according to new methods of inquiry. The certainty of 


the conclusions in geometry and arithmetic brought out in his 
mind the contrast between the true and false ways of seeking 
the truth. He thereupon attempted to apply mathematical 
reasoning to all sciences. "Comparing the mysteries of nature 
with the laws of mathematics, he dared to hope that the 
secrets of both could be unlocked with the same key." Thus 
he built up a system of philosophy called Cartesianism. 

Great as was Descartes' celebrity as a metaphysician, it may 
be fairly questioned whether his claim to be remembered by 
posterity as a mathematician is not greater. His philosophy 
has long since been superseded by other systems, but the 
analytical geometry of Descartes will remain a valuable 
possession forever. At the age of twenty-one, Descartes 
enlisted in the army of Prince Maurice of Orange. His years of 
soldiering were years of leisure, in which he had time to pursue 
his studies. At that time mathematics was his favourite 
science. But in 1625 he ceased to devote himself to pure 
mathematics. Sir William Hamilton is in error when he states 
that Descartes considered mathematical studies absolutely 
pernicious as a means of internal culture. In a letter to 
Mersenne, Descartes says: "M. Desargues puts me under 
obligations on account of the pains that it has pleased him 
to have in me, in that he shows that he is sorry that I do 
not wish to study more in geometry, but I have resolved to 
quit only abstract geometry, that is to say, the consideration 
of questions which serve only to exercise the mind, and this, 
in order to study another kind of geometry, which has for its 
object the explanation of the phenomena of nature. . . . You 


know that all my physics is nothing else than geometry" The 
years between 1629 and 1649 were passed by him in Holland 
in the study, principally, of physics and metaphysics. His 
residence in Holland was during the most brilliant days of 
the Dutch state. In 1637 he published his Discours de la 
Methode, containing among others an essay of 106 pages on 
geometry. His Geometry is not easy reading. An edition 
appeared subsequently with notes by his friend De Beaune, 
which were intended to remove the difficulties. 

It is frequently stated that Descartes was the first to 
apply algebra to geometry. This statement is inaccurate, 
for Vieta and others had done this before him. Even the 
Arabs sometimes used algebra in connection with geometry. 
The new step that Descartes did take was the introduction 
into geometry of an analytical method based on the notion 
of variables and constants, which enabled him to represent 
curves by algebraic equations. In the Greek geometry, the idea 
of motion was wanting, but with Descartes it became a very 
fruitful conception. By him a point on a plane was determined 
in position by its distances from two fixed right lines or axes. 
These distances varied with every change of position in the 
point. This geometric idea of co-ordinate representation, 
together with the algebraic idea of two variables in one 
equation having an indefinite number of simultaneous values, 
furnished a method for the study of loci, which is admirable 
for the generality of its solutions. Thus the entire conic 
sections of Apollonius is wrapped up and contained in a single 
equation of the second degree. 


The Latin term for "ordinate" used by Descartes comes 
from the expression linecB ordinate, employed by Roman 
surveyors for parallel lines. The term abscissa occurs for the 
first time in a Latin work of 1659, written by Stefano degli 
Angeli (1623-1697), a professor of mathematics in Rome. [3] 
Descartes' geometry was called "analytical geometry," partly 
because, unlike the synthetic geometry of the ancients, it is 
actually analytical, in the sense that the word is used in logic; 
and partly because the practice had then already arisen, of 
designating by the term analysis the calculus with general 

The first important example solved by Descartes in his 
geometry is the "problem of Pappus"; viz. "Given several 
straight lines in a plane, to find the locus of a point such that 
the perpendiculars, or more generally, straight lines at given 
angles, drawn from the point to the given lines, shall satisfy 
the condition that the product of certain of them shall be in 
a given ratio to the product of the rest." Of this celebrated 
problem, the Greeks solved only the special case when the 
number of given lines is four, in which case the locus of the 
point turns out to be a conic section. By Descartes it was 
solved completely, and it afforded an excellent example of the 
use which can be made of his analytical method in the study 
of loci. Another solution was given later by Newton in the 

The methods of drawing tangents invented by Roberval 
and Fermat were noticed earlier. Descartes gave a third 
method. Of all the problems which he solved by his geometry, 


none gave him as great pleasure as his mode of constructing 
tangents. It is profound but operose, and, on that account, 
inferior to Fermat's. His solution rests on the method of 
Indeterminate Coefficients, of which he bears the honour of 
invention. Indeterminate coefficients were employed by him 
also in solving biquadratic equations. 

The essays of Descartes on dioptrics and geometry were 
sharply criticised by Fermat, who wrote objections to the 
former, and sent his own treatise on "maxima and minima" 
to show that there were omissions in the geometry. Descartes 
thereupon made an attack on Fermat's method of tangents. 
Descartes was in the wrong in this attack, yet he continued 
the controversy with obstinacy. He had a controversy also 
with Roberval on the cycloid. This curve has been called the 
"Helen of geometers," on account of its beautiful properties 
and the controversies which their discovery occasioned. Its 
quadrature by Roberval was generally considered a brilliant 
achievement, but Descartes commented on it by saying that 
any one moderately well versed in geometry might have 
done this. He then sent a short demonstration of his own. 
On Roberval's intimating that he had been assisted by a 
knowledge of the solution, Descartes constructed the tangent 
to the curve, and challenged Roberval and Fermat to do the 
same. Fermat accomplished it, but Roberval never succeeded 
in solving this problem, which had cost the genius of Descartes 
but a moderate degree of attention. 

He studied some new curves, now called "ovals of Descartes," 
which were intended by him to serve in the construction of 


converging lenses, but which yielded no results of practical 

The application of algebra to the doctrine of curved lines 
reacted favourably upon algebra. As an abstract science, 
Descartes improved it by the systematic use of exponents 
and by the full interpretation and construction of negative 
quantities. Descartes also established some theorems on the 
theory of equations. Celebrated is his "rule of signs" for 
determining the number of positive and negative roots; viz. an 
equation may have as many + roots as there are variations 
of signs, and as many — roots as there are permanencies of 
signs. Descartes was charged by Wallis with availing himself, 
without acknowledgment, of Harriot's theory of equations, 
particularly his mode of generating equations; but there seems 
to be no good ground for the charge. Wallis also claimed 
that Descartes failed to observe that the above rule of signs 
is not true whenever the equation has imaginary roots; but 
Descartes does not say that the equation always has, but that 
it may have so many roots. It is true that Descartes does 
not consider the case of imaginaries directly, but further on in 
his Geometry he gives incontestable evidence of being able to 
handle this case also. 

In mechanics, Descartes can hardly be said to have advanced 
beyond Galileo. The latter had overthrown the ideas of 
Aristotle on this subject, and Descartes simply "threw himself 
upon the enemy" that had already been "put to the rout." 
His statement of the first and second laws of motion was an 
improvement in form, but his third law is false in substance. 


The motions of bodies in their direct impact was imperfectly 
understood by Galileo, erroneously given by Descartes, and 
first correctly stated by Wren, Wallis, and Huygens. 

One of the most devoted pupils of Descartes was the learned 
Princess Elizabeth, daughter of Frederick V. She applied the 
new analytical geometry to the solution of the "Apollonian 
problem." His second royal follower was Queen Christina, the 
daughter of Gustavus Adolphus. She urged upon Descartes to 
come to the Swedish court. After much hesitation he accepted 
the invitation in 1649. He died at Stockholm one year later. 
His life had been one long warfare against the prejudices of 

It is most remarkable that the mathematics and philoso- 
phy of Descartes should at first have been appreciated less 
by his countrymen than by foreigners. The indiscreet tem- 
per of Descartes alienated the great contemporary French 
mathematicians, Roberval, Fermat, Pascal. They continued 
in investigations of their own, and on some points strongly 
opposed Descartes. The universities of France were under 
strict ecclesiastical control and did nothing to introduce his 
mathematics and philosophy. It was in the youthful universi- 
ties of Holland that the effect of Cartesian teachings was most 
immediate and strongest. 

The only prominent Frenchman who immediately followed 
in the footsteps of the great master was De Beaune (1601- 
1652). He was one of the first to point out that the properties 
of a curve can be deduced from the properties of its tangent. 
This mode of inquiry has been called the inverse method 


of tangents. He contributed to the theory of equations by 
considering for the first time the upper and lower limits of the 
roots of numerical equations. 

In the Netherlands a large number of distinguished math- 
ematicians were at once struck with admiration for the 
Cartesian geometry. Foremost among these are van Schooten, 
John de Witt, van Heuraet, Sluze, and Hudde. Van Schooten 
(died 1660), professor of mathematics at Ley den, brought out 
an edition of Descartes' geometry, together with the notes 
thereon by De Beaune. His chief work is his Exercitationes 
Mathematics, in which he applies the analytical geometry to 
the solution of many interesting and difficult problems. The 
noble- hearted Johann de Witt, grand-pensioner of Holland, 
celebrated as a statesman and for his tragical end, was an 
ardent geometrician. He conceived a new and ingenious way 
of generating conies, which is essentially the same as that 
by projective pencils of rays in modern synthetic geometry. 
He treated the subject not synthetically, but with aid of the 
Cartesian analysis. Rene Frangois de Sluze (1622-1685) 
and Johann Hudde (1633-1704) made some improvements 
on Descartes' and Fermat's methods of drawing tangents, and 
on the theory of maxima and minima. With Hudde, we find 
the first use of three variables in analytical geometry. He is 
the author of an ingenious rule for finding equal roots. We 
illustrate it by the equation x 3 — x 2 — 8x + 12 = 0. Taking an 
arithmetical progression 3, 2, 1, 0, of which the highest term is 
equal to the degree of the equation, and multiplying each term 
of the equation respectively by the corresponding term of the 


progression, we get 3x 3 — 2x 2 — 8x = 0, or 3x 2 — 2x — 8 = 0. 
This last equation is by one degree lower than the original one. 
Find the G.C.D. of the two equations. This is x — 2; hence 
2 is one of the two equal roots. Had there been no common 
divisor, then the original equation would not have possessed 
equal roots. Hudde gave a demonstration for this rule. [24] 

Heinrich van Heuraet must be mentioned as one of the 
earliest geometers who occupied themselves with success in 
the rectification of curves. He observed in a general way 
that the two problems of quadrature and of rectification are 
really identical, and that the one can be reduced to the other. 
Thus he carried the rectification of the hyperbola back to 
the quadrature of the hyperbola. The semi-cubical parabola 
y 3 = ax 2 was the first curve that was ever rectified absolutely. 
This appears to have been accomplished independently by 
Van Heuraet in Holland and by William Neil (1637-1670) 
in England. According to Wallis the priority belongs to Neil. 
Soon after, the cycloid was rectified by Wren and Fermat. 

The prince of philosophers in Holland, and one of the 
greatest scientists of the seventeenth century, was Christian 
Huygens (1629-1695), a native of the Hague. Eminent as a 
physicist and astronomer, as well as mathematician, he was 
a worthy predecessor of Sir Isaac Newton. He studied at 
Ley den under the younger Van Schooten. The perusal of some 
of his earliest theorems led Descartes to predict his future 
greatness. In 1651 Huygens wrote a treatise in which he 
pointed out the fallacies of Gregory St. Vincent (1584-1667) 
on the subject of quadratures. He himself gave a remarkably 


close and convenient approximation to the length of a circular 
arc. In 1660 and 1663 he went to Paris and to London. In 
1666 he was appointed by Louis XIV. member of the French 
Academy of Sciences. He was induced to remain in Paris 
from that time until 1681, when he returned to his native city, 
partly for consideration of his health and partly on account of 
the revocation of the Edict of Nantes. 

The majority of his profound discoveries were made with 
aid of the ancient geometry, though at times he used the 
geometry of Descartes or of Cavalieri and Fermat. Thus, 
like his illustrious friend, Sir Isaac Newton, he always showed 
partiality for the Greek geometry. Newton and Huygens 
were kindred minds, and had the greatest admiration for 
each other. Newton always speaks of him as the "Summus 

To the two curves (cubical parabola and cycloid) previously 
rectified he added a third, — the cissoid. He solved the problem 
of the catenary, determined the surface of the parabolic and 
hyperbolic conoid, and discovered the properties of the 
logarithmic curve and the solids generated by it. Huygens' De 
horologio oscillatorio (Paris, 1673) is a work that ranks second 
only to the Principia of Newton and constitutes historically 
a necessary introduction to it. [13] The book opens with 
a description of pendulum clocks, of which Huygens is the 
inventor. Then follows a treatment of accelerated motion 
of bodies falling free, or sliding on inclined planes, or on 
given curves, — culminating in the brilliant discovery that the 
cycloid is the tautochronous curve. To the theory of curves he 


added the important theory of "evolutes." After explaining 
that the tangent of the evolute is normal to the involute, 
he applied the theory to the cycloid, and showed by simple 
reasoning that the evolute of this curve is an equal cycloid. 
Then comes the complete general discussion of the centre of 
oscillation. This subject had been proposed for investigation 
by Mersenne and discussed by Descartes and Roberval. In 
Huygens' assumption that the common centre of gravity of a 
group of bodies, oscillating about a horizontal axis, rises to its 
original height, but no higher, is expressed for the first time 
one of the most beautiful principles of dynamics, afterwards 
called the principle of the conservation of vis viva. [32] The 
thirteen theorems at the close of the work relate to the theory 
of centrifugal force in circular motion. This theory aided 
Newton in discovering the law of gravitation. 

Huygens wrote the first formal treatise on probability. He 
proposed the wave-theory of light and with great skill applied 
geometry to its development. This theory was long neglected, 
but was revived and successfully worked out by Young and 
Fresnel a century later. Huygens and his brother improved the 
telescope by devising a better way of grinding and polishing 
lenses. With more efficient instruments he determined the 
nature of Saturn's appendage and solved other astronomical 
questions. Huygens' Opuscula posthuma appeared in 1703. 

Passing now from Holland to England, we meet there one of 
the most original mathematicians of his day — John Wallis 
(1616-1703). He was educated for the Church at Cambridge 
and entered Holy Orders. But his genius was employed chiefly 


in the study of mathematics. In 1649 he was appointed 
Savilian professor of geometry at Oxford. He was one of the 
original members of the Royal Society, which was founded 
in 1663. Wallis thoroughly grasped the mathematical methods 
both of Cavalieri and Descartes. His Conic Sections is the 
earliest work in which these curves are no longer considered as 
sections of a cone, but as curves of the second degree, and are 
treated analytically by the Cartesian method of co-ordinates. 
In this work Wallis speaks of Descartes in the highest terms, 
but in his Algebra he, without good reason, accuses Descartes 
of plagiarising from Harriot. We have already mentioned 
elsewhere Wallis's solution of the prize questions on the 
cycloid, which were proposed by Pascal. 

The Arithmetic of Infinites, published in 1655, is his 
greatest work. By the application of analysis to the Method of 
Indivisibles, he greatly increased the power of this instrument 
for effecting quadratures. He advanced beyond Kepler by 
making more extended use of the "law of continuity" and 
placing full reliance in it. By this law he was led to regard the 
denominators effractions as powers with negative exponents. 
Thus, the descending geometrical progression x 3 , x 2 , x , x°, if 
continued, gives x _1 , x -2 , x -3 , etc.; which is the same thing 
as -, — s-, — s-. The exponents of this geometric series are in 
continued arithmetical progression, 3, 2, 1, 0, — 1, —2, —3. 
He also used fractional exponents, which, like the negative, 
had been invented long before, but had failed to be generally 
introduced. The symbol oo for infinity is due to him. 

Cavalieri and the French geometers had ascertained the 



formula for squaring the parabola of any degree, y = x 
m being a positive integer. By the summation of the powers 
of the terms of infinite arithmetical series, it was found that 
the curve y = x m is to the area of the parallelogram having 
the same base and altitude as 1 is to m + 1. Aided by the law 
of continuity, Wallis arrived at the result that this formula 
holds true not only when m is positive and integral, but 
also when it is fractional or negative. Thus, in the parabola 
y = \ppx, m = ^ ; hence the area of the parabolic segment is 
to that of the circumscribed rectangle as 1 : 1^, or as 2 : 3. 
Again, suppose that in y = x m , m = — \\ then the curve 
is a kind of hyperbola referred to its asymptotes, and the 
hyperbolic space between the curve and its asymptotes is to 
the corresponding parallelogram as 1 : \. If m = —1, as in the 
common equilateral hyperbola y = x~ l or xy = 1, then this 
ratio is 1 : —1 + 1, or 1 : 0, showing that its asymptotic space 
is infinite. But in the case when m is greater than unity and 
negative, Wallis was unable to interpret correctly his results. 
For example, if m = —3, then the ratio becomes 1 : —2, or as 
unity to a negative number. What is the meaning of this? 
Wallis reasoned thus: If the denominator is only zero, then 
the area is already infinite; but if it is less than zero, then the 
area must be more than infinite. It was pointed out later by 
Varignon, that this space, supposed to exceed infinity, is really 
finite, but taken negatively; that is, measured in a contrary 
direction. [31] The method of Wallis was easily extended to 

m P 

cases such as y = ax "n + bx i by performing the quadrature for 
each term separately, and then adding the results. 


The manner in which Wallis studied the quadrature of 
the circle and arrived at his expression for the value of it is 
extraordinary. He found that the areas comprised between 
the axes, the ordinate corresponding to x, and the curves 
represented by the equations y = (1 — x 2 ) , y = (1 — x 2 ) , 
y = (1 — x 2 ) 2 , y = (1 — x 2 ) 3 , etc., are expressed in functions of 
the circumscribed rectangles having x and y for their sides, by 
the quantities forming the series 



X O X 

X — <rX -j- irX , 

™ O ^O I O „Q l^i «-/-/■» 

X 7>X "J" ^X Tjjy j Ctt. 

When x = 1, these values become respectively 1, |, ^, yj|j, 
etc. Now since the ordinate of the circle is y = (1 — x 2 )2, 
the exponent of which is ^ or the mean value between 
and 1, the question of this quadrature reduced itself to this: 
If 0, 1, 2, 3, etc., operated upon by a certain law, give 
1, |, yjj, y[|j, what will ^ give, when operated upon by the 
same law? He attempted to solve this by interpolation, a 
method first brought into prominence by him, and arrived by 
a highly complicated and difficult analysis at the following 
very remarkable expression: 

7T 2-2-4-4-6-6-8-8--- 

2 1-3-3-5-5-7-7-9- 

He did not succeed in making the interpolation itself, 
because he did not employ literal or general exponents, and 


could not conceive a series with more than one term and less 
than two, which it seemed to him the interpolated series must 
have. The consideration of this difficulty led Newton to the 
discovery of the Binomial Theorem. This is the best place 
to speak of that discovery. Newton virtually assumed that 
the same conditions which underlie the general expressions 
for the areas given above must also hold for the expression to 
be interpolated. In the first place, he observed that in each 
expression the first term is x, that x increases in odd powers, 
that the signs alternate + and — , and that the second terms 
jjx 3 , jx 3 , |x 3 , |x 3 , are in arithmetical progression. Hence 

T rr ,o 

the first two terms of the interpolated series must be x — - — . 

He next considered that the denominators 1, 3, 5, 7, etc., are 

in arithmetical progression, and that the coefficients in the 

numerators in each expression are the digits of some power 

of the number 11; namely, for the first expression, 11° or 1; 

for the second, ll 1 or 1, 1; for the third, ll 2 or 1, 2, 1; for 

the fourth, ll 3 or 1, 3, 3, 1; etc. He then discovered that, 

having given the second digit (call it m), the remaining digits 

can be found by continual multiplication of the terms of the 
m-0 m-1 m-2 m-3 _,, .„ . 

series etc. ihus, it m = 4, then 

12 3 4 
m — 1 . m — 2. m — 3 . . . . 

4 gives 6; 6 gives 4; 4 gives 1. Applying 

2 3 4 ±x 3 

this rule to the required series, since the second term is - — , 

we have m — \, and then get for the succeeding coefficients 
in the numerators respectively — |, + jg, — j|§, etc.; hence the 

required area for the circular segment is x — — 

o o ( 


etc. Thus he found the interpolated expression to be an infinite 
series, instead of one having more than one term and less 
than two, as Wallis believed it must be. This interpolation 
suggested to Newton a mode of expanding (1 — x 1 ) 2 , or, more 
generally, (1 — x 2 ) m , into a series. He observed that he had 
only to omit from the expression just found the denominators 
1, 3, 5, 7, etc., and to lower each power of x by unity, and he 
had the desired expression. In a letter to Oldenburg (June 13, 
1676), Newton states the theorem as follows: The extraction 
of roots is much shortened by the theorem 

(P + PQ) » = Pn + —AQ + — BQ + — CQ + etc., 

n 2n 6n 

where A means the first term, Pn, B the second term, C 
the third term, etc. He verified it by actual multiplication, 
but gave no regular proof of it. He gave it for any exponent 
whatever, but made no distinction between the case when the 
exponent is positive and integral, and the others. 

It should here be mentioned that very rude beginnings of 
the binomial theorem are found very early. The Hindoos and 
Arabs used the expansions of (a + b) 2 and (0+ 6) 3 for extracting 
roots; Vieta knew the expansion of (a + 6) 4 ; but these were the 
results of simple multiplication without the discovery of any 
law. The binomial coefficients for positive whole exponents 
were known to some Arabic and European mathematicians. 
Pascal derived the coefficients from the method of what is 
called the "arithmetical triangle." Lucas de Burgo, Stifel, 
Stevinus, Briggs, and others, all possessed something from 
which one would think the binomial theorem could have been 


gotten with a little attention, "if we did not know that such 
simple relations were difficult to discover." 

Though Wallis had obtained an entirely new expression 
for 7r, he was not satisfied with it; for instead of a finite number 
of terms yielding an absolute value, it contained merely an 
infinite number, approaching nearer and nearer to that value. 
He therefore induced his friend, Lord Brouncker (1620?- 
1684), the first president of the Royal Society, to investigate 
this subject. Of course Lord Brouncker did not find what they 
were after, but he obtained the following beautiful equality:— 


IT = 


1 + 

2+ — 9 




2 + 

2 + etc. 

Continued fractions, both ascending and descending, appear 
to have been known already to the Greeks and Hindoos, 
though not in our present notation. Brouncker's expression 
gave birth to the theory of continued fractions. 

Wallis' method of quadratures was diligently studied by his 
disciples. Lord Brouncker obtained the first infinite series for 
the area of an equilateral hyperbola between its asymptotes. 
Nicolaus Mercator of Holstein, who had settled in England, 
gave, in his Logarithmotechnia (London, 1668), a similar 
series. He started with the grand property of the equilateral 
hyperbola, discovered in 1647 by Gregory St. Vincent, which 
connected the hyperbolic space between the asymptotes with 


the natural logarithms and led to these logarithms being called 
hyperbolic. By it Mercator arrived at the logarithmic series, 
which Wallis had attempted but failed to obtain. He showed 
how the construction of logarithmic tables could be reduced 
to the quadrature of hyperbolic spaces. Following up some 
suggestions of Wallis, William Neil succeeded in rectifying 
the cubical parabola, and Wren in rectifying any cycloidal 

A prominent English mathematician and contemporary of 
Wallis was Isaac Barrow (1630-1677). He was professor of 
mathematics in London, and then in Cambridge, but in 1669 
he resigned his chair to his illustrious pupil, Isaac Newton, 
and renounced the study of mathematics for that of divinity. 
As a mathematician, he is most celebrated for his method of 
tangents. He simplified the method of Fermat by introducing 
two infinitesimals instead of one, and approximated to the 
course of reasoning afterwards followed by Newton in his 
doctrine on Ultimate Ratios. 

He considered the infinitesimal right triangle ABB' having 

for its sides the difference 
between two successive ordi- 
nates, the distance between 
them, and the portion of the 
curve intercepted by them. 
This triangle is similar to 
BPT, formed by the ordi- 
nate, the tangent, and the 
sub-tangent. Hence, if we know the ratio of B'A to BA, then 


we know the ratio of the ordinate and the sub-tangent, and 
the tangent can be constructed at once. For any curve, say 
y 2 = px, the ratio of B'A to BA is determined from its equation 
as follows: If x receives an infinitesimal increment PP' = e, 
then y receives an increment B'A = a, and the equation for 
the ordinate B'P' becomes y 2 + 2ay + a 2 = px + pe. Since 
y 2 = px, we get 2ay + a 2 = pe; neglecting higher powers of the 
infinitesimals, we have lay = pe, which gives 

a : e = p : 2y = p : 2 v / px. 

But a : e = the ordinate : the sub-tangent; hence 

p : 2 x /px = ^Jpx : sub-tangent, 

giving 2x for the value of the sub-tangent. This method differs 
from that of the differential calculus only in notation. [31] 


It has been seen that in France prodigious scientific progress 
was made during the beginning and middle of the seventeenth 
century. The toleration which marked the reign of Henry 
IV. and Louis XIII. was accompanied by intense intellectual 
activity. Extraordinary confidence came to be placed in the 
power of the human mind. The bold intellectual conquests 
of Descartes, Fermat, and Pascal enriched mathematics with 
imperishable treasures. During the early part of the reign of 
Louis XIV. we behold the sunset splendour of this glorious 
period. Then followed a night of mental effeminacy. This 


lack of great scientific thinkers during the reign of Louis XIV. 
may be due to the simple fact that no great minds were born; 
but, according to Buckle, it was due to the paternalism, to 
the spirit of dependence and subordination, and to the lack of 
toleration, which marked the policy of Louis XIV. 

In the absence of great French thinkers, Louis XIV. sur- 
rounded himself by eminent foreigners. Romer from Denmark, 
Huygens from Holland, Dominic Cassini from Italy, were the 
mathematicians and astronomers adorning his court. They 
were in possession of a brilliant reputation before going to 
Paris. Simply because they performed scientific work in Paris, 
that work belongs no more to France than the discoveries of 
Descartes belong to Holland, or those of Lagrange to Germany, 
or those of Euler and Poncelet to Russia. We must look to 
other countries than France for the great scientific men of the 
latter part of the seventeenth century. 

About the time when Louis XIV. assumed the direction of 
the French government Charles II. became king of England. 
At this time England was extending her commerce and navi- 
gation, and advancing considerably in material prosperity. A 
strong intellectual movement took place, which was unwit- 
tingly supported by the king. The age of poetry was soon 
followed by an age of science and philosophy. In two successive 
centuries England produced Shakespeare and Newton! 

Germany still continued in a state of national degradation. 
The Thirty Years' War had dismembered the empire and 
brutalised the people. Yet this darkest period of Germany's 
history produced Leibniz, one of the greatest geniuses of 


modern times. 

There are certain focal points in history toward which the 

lines of past progress converge, and from which radiate the 

advances of the future. Such was the age of Newton and Leibniz 

in the history of mathematics. During fifty years preceding 

this era several of the brightest and acutest mathematicians 

bent the force of their genius in a direction which finally led 

to the discovery of the infinitesimal calculus by Newton and 

Leibniz. Cavalieri, Roberval, Fermat, Descartes, Wallis, and 

others had each contributed to the new geometry. So great 

was the advance made, and so near was their approach toward 

the invention of the infinitesimal analysis, that both Lagrange 

and Laplace pronounced their countryman, Fermat, to be the 

true inventor of it. The differential calculus, therefore, was 

not so much an individual discovery as the grand result of a 

succession of discoveries by different minds. Indeed, no great 

discovery ever flashed upon the mind at once, and though 

those of Newton will influence mankind to the end of the 

world, yet it must be admitted that Pope's lines are only a 

"poetic fancy" :— 

"Nature and Nature's laws lay hid in night; 
God said, 'Let Newton be,' and all was light." 

Isaac Newton (1642-1727) was born at Woolsthorpe, in 
Lincolnshire, the same year in which Galileo died. At his 
birth he was so small and weak that his life was despaired 
of. His mother sent him at an early age to a village school, 
and in his twelfth year to the public school at Grantham. At 
first he seems to have been very inattentive to his studies and 


very low in the school; but when, one day, the little Isaac 
received a severe kick upon his stomach from a boy who was 
above him, he laboured hard till he ranked higher in school 
than his antagonist. From that time he continued to rise 
until he was the head boy. [33] At Grantham, Isaac showed 
a decided taste for mechanical inventions. He constructed 
a water-clock, a wind-mill, a carriage moved by the person 
who sat in it, and other toys. When he had attained his 
fifteenth year his mother took him home to assist her in the 
management of the farm, but his great dislike for farm-work 
and his irresistible passion for study, induced her to send 
him back to Grantham, where he remained till his eighteenth 
year, when he entered Trinity College, Cambridge (1660). 
Cambridge was the real birthplace of Newton's genius. Some 
idea of his strong intuitive powers may be drawn from the 
fact that he regarded the theorems of ancient geometry 
as self-evident truths, and that, without any preliminary 
study, he made himself master of Descartes' Geometry. He 
afterwards regarded this neglect of elementary geometry a 
mistake in his mathematical studies, and he expressed to Dr. 
Pemberton his regret that "he had applied himself to the 
works of Descartes and other algebraic writers before he had 
considered the Elements of Euclid with that attention which 
so excellent a writer deserves." Besides Descartes' Geometry, 
he studied Oughtred's Clavis, Kepler's Optics, the works of 
Vieta, Schooten's Miscellanies, Barrow's Lectures, and the 
works of Wallis. He was particularly delighted with Wallis' 
Arithmetic of Infinites, a treatise fraught with rich and varied 


suggestions. Newton had the good fortune of having for 
a teacher and fast friend the celebrated Dr. Barrow, who 
had been elected professor of Greek in 1660, and was made 
Lucasian professor of mathematics in 1663. The mathematics 
of Barrow and of Wallis were the starting-points from which 
Newton, with a higher power than his masters', moved onward 
into wider fields. Wallis had effected the quadrature of curves 
whose ordinates are expressed by any integral and positive 
power of (1 — x 2 ). We have seen how Wallis attempted 
but failed to interpolate between the areas thus calculated, 
the areas of other curves, such as that of the circle; how 
Newton attacked the problem, effected the interpolation, and 
discovered the Binomial Theorem, which afforded a much 
easier and direct access to the quadrature of curves than did 
the method of interpolation; for even though the binomial 
expression for the ordinate be raised to a fractional or negative 
power, the binomial could at once be expanded into a series, 
and the quadrature of each separate term of that series could 
be effected by the method of Wallis. Newton introduced the 
system of literal indices. 

Newton's study of quadratures soon led him to another and 
most profound invention. He himself says that in 1665 and 
1666 he conceived the method of fluxions and applied them to 
the quadrature of curves. Newton did not communicate the 
invention to any of his friends till 1669, when he placed in the 
hands of Barrow a tract, entitled De Analyst per Mquationes 
Numero Terminorum Infinitas, which was sent by Barrow to 
Collins, who greatly admired it. In this treatise the principle 


of fluxions, though distinctly pointed out, is only partially 
developed and explained. Supposing the abscissa to increase 
uniformly in proportion to the time, he looked upon the 
area of a curve as a nascent quantity increasing by continued 
fluxion in the proportion of the length of the ordinate. The 
expression which was obtained for the fluxion he expanded 
into a finite or infinite series of monomial terms, to which 
Wallis' rule was applicable. Barrow urged Newton to publish 
this treatise; "but the modesty of the author, of which the 
excess, if not culpable, was certainly in the present instance 
very unfortunate, prevented his compliance." [26] Had this 
tract been published then, instead of forty-two years later, 
there would probably have been no occasion for that long and 
deplorable controversy between Newton and Leibniz. 

For a long time Newton's method remained unknown, 
except to his friends and their correspondents. In a letter 
to Collins, dated December 10th, 1672, Newton states the 
fact of his invention with one example, and then says: "This 
is one particular, or rather corollary, of a general method, 
which extends itself, without any troublesome calculation, 
not only to the drawing of tangents to any curve lines, whether 
geometrical or mechanical, or anyhow respecting right lines or 
other curves, but also to the resolving other abstruser kinds 
of problems about the crookedness, areas, lengths, centres 
of gravity of curves, etc.; nor is it (as Hudden's method of 
Maximis and Minimis) limited to equations which are free 
from surd quantities. This method I have interwoven with 
that other of working in equations, by reducing them to 


infinite series." 

These last words relate to a treatise he composed in the 
year 1671, entitled Method of Fluxions, in which he aimed 
to represent his method as an independent calculus and as a 
complete system. This tract was intended as an introduction to 
an edition of Kinckhuysen's Algebra, which he had undertaken 
to publish. "But the fear of being involved in disputes about 
this new discovery, or perhaps the wish to render it more 
complete, or to have the sole advantage of employing it in his 
physical researches, induced him to abandon this design." [33] 

Excepting two papers on optics, all of his works appear to 
have been published only after the most pressing solicitations 
of his friends and against his own wishes. [34] His researches 
on light were severely criticised, and he wrote in 1675: "I 
was so persecuted with discussions arising out of my theory 
of light that I blamed my own imprudence for parting with so 
substantial a blessing as my quiet to run after a shadow." 

The Method of Fluxions, translated by J. Colson from 
Newton's Latin, was first published in 1736, or sixty-five years 
after it was written. In it he explains first the expansion 
into series of fractional and irrational quantities, — a subject 
which, in his first years of study, received the most careful 
attention. He then proceeds to the solution of the two 
following mechanical problems, which constitute the pillars, 
so to speak, of the abstract calculus: — 

"I. The length of the space described being continually 
(i.e. at all times) given; to find the velocity of the motion at 
any time proposed. 


"II. The velocity of the motion being continually given; to 
find the length of the space described at any time proposed." 

Preparatory to the solution, Newton says: "Thus, in the 
equation y = x 2 , if y represents the length of the space at any 
time described, which (time) another space x, by increasing 
with an uniform celerity x, measures and exhibits as described: 
then 2xx will represent the celerity by which the space y, at 
the same moment of time, proceeds to be described; and 
contrary wise." 

"But whereas we need not consider the time here, any 
farther than it is expounded and measured by an equable 
local motion; and besides, whereas only quantities of the same 
kind can be compared together, and also their velocities of 
increase and decrease; therefore, in what follows I shall have 
no regard to time formally considered, but I shall suppose 
some one of the quantities proposed, being of the same kind, 
to be increased by an equable fluxion, to which the rest may be 
referred, as it were to time; and, therefore, by way of analogy, 
it may not improperly receive the name of time." In this 
statement of Newton there is contained a satisfactory answer 
to the objection which has been raised against his method, 
that it introduces into analysis the foreign idea of motion. A 
quantity thus increasing by uniform fluxion, is what we now 
call an independent variable. 

Newton continues: "Now those quantities which I consider 
as gradually and indefinitely increasing, I shall hereafter call 
fluents, or flowing quantities, and shall represent them by 
the final letters of the alphabet, v, x, y, and z; . . . and the 


velocities by which every fluent is increased by its generating 
motion (which I may call fluxions, or simply velocities, or 
celerities), I shall represent by the same letters pointed, thus, 
v, x, y, z. That is, for the celerity of the quantity v I shall put v, 
and so for the celerities of the other quantities x, y, and z, I 
shall put x, y, and z, respectively." It must here be observed 
that Newton does not take the fluxions themselves infinitely 
small. The "moments of fluxions," a term introduced further 
on, are infinitely small quantities. These "moments," as 
defined and used in the Method of Fluxions, are substantially 
the differentials of Leibniz. De Morgan points out that no 
small amount of confusion has arisen from the use of the 
word fluxion and the notation x by all the English writers 
previous to 1704, excepting Newton and Cheyne, in the sense 
of an infinitely small increment. [35] Strange to say, even in 
the Commercium Epistolicum the words moment and fluxion 
appear to be used as synonymous. 

After showing by examples how to solve the first problem, 
Newton proceeds to the demonstration of his solution:— 

"The moments of flowing quantities (that is, their indef- 
initely small parts, by the accession of which, in infinitely 
small portions of time, they are continually increased) are as 
the velocities of their flowing or increasing. 

"Wherefore, if the moment of any one (as x) be represented 
by the product of its celerity x into an infinitely small 
quantity (i.e. by xO), the moments of the others, v, y, z, will 
be represented by vO, yO, iO; because vO, xO, yO, and iO are to 
each other as v, x, y, and z. 


"Now since the moments, as xO and yO, are the indefinitely 
little accessions of the flowing quantities x and y, by which 
those quantities are increased through the several indefinitely 
little intervals of time, it follows that those quantities, x and y, 
after any indefinitely small interval of time, become x + xO 
and y + yO, and therefore the equation, which at all times 
indifferently expresses the relation of the flowing quantities, 
will as well express the relation between x + xO and y + yO, as 
between x and y; so that x + xO and y + yO may be substituted 
in the same equation for those quantities, instead of x and y. 
Thus let any equation x 3 — ax 2 + axy — y 3 = be given, and 
substitute x + xO for x, and y + yO for y, and there will arise 

x 3, + 3x 2 x0 + 3xx0x0 + x 3 3 ' 

-ax — 2axx0 — axOxO 

+axy + ayxO + axOyO 


+ axyO 
-y 3 - 3y 2 ij0 - SyyOyO - y 3 3 

"Now, by supposition, x 3 — ax 2 + axy — y 3 = 0, which 
therefore, being expunged and the remaining terms being 
divided by 0, there will remain 

2 2 

3x x — 2axx + ayx + axy — 3y y + 3xxx0 — axxO + axyO 

- 3yyy0 + x 3 00 - y 3 00 = 0. 

But whereas zero is supposed to be infinitely little, that it 
may represent the moments of quantities, the terms that are 
multiplied by it will be nothing in respect of the rest (termini 


in earn ducti pro nihilo possunt haberi cum aliis collati); 
therefore I reject them, and there remains 

3x x — 2axx + ayx + axy — 3y y = 0, 

as above in Example I." Newton here uses infinitesimals. 

Much greater than in the first problem were the difficulties 
encountered in the solution of the second problem, involving, 
as it does, inverse operations which have been taxing the skill 
of the best analysts since his time. Newton gives first a special 
solution to the second problem in which he resorts to a rule 
for which he has given no proof. 

In the general solution of his second problem, Newton 
assumed homogeneity with respect to the fluxions and then 
considered three cases: (1) when the equation contains two 
fluxions of quantities and but one of the fluents; (2) when 
the equation involves both the fluents as well as both the 
fluxions; (3) when the equation contains the fluents and the 
fluxions of three or more quantities. The first case is the 


easiest since it requires simply the integration of — = f(x), 

to which his "special solution" is applicable. The second 

case demanded nothing less than the general solution of a 

differential equation of the first order. Those who know what 

efforts were afterwards needed for the complete exploration 

of this field in analysis, will not depreciate Newton's work 

even though he resorted to solutions in form of infinite series. 

Newton's third case comes now under the solution of partial 

differential equations. He took the equation 2x — z + xy = 

and succeeded in finding a particular integral of it. 


The rest of the treatise is devoted to the determination of 
maxima and minima, the radius of curvature of curves, and 
other geometrical applications of his fluxionary calculus. All 
this was done previous to the year 1672. 

It must be observed that in the Method of Fluxions (as 
well as in his De Analyst and all earlier papers) the method 
employed by Newton is strictly infinitesimal, and in substance 
like that of Leibniz. Thus, the original conception of the 
calculus in England, as well as on the Continent, was based on 
infinitesimals. The fundamental principles of the fluxionary 
calculus were first given to the world in the Principia; but its 
peculiar notation did not appear until published in the second 
volume of Wallis' Algebra in 1693. The exposition given in 
the Algebra was substantially a contribution of Newton; it 
rests on infinitesimals. In the first edition of the Principia 
(1687) the description of fluxions is likewise founded on 
infinitesimals, but in the second (1713) the foundation is 
somewhat altered. In Book II. Lemma II. of the first 
edition we read: "Cave tamen intellexeris particulas finitas. 
Momenta quam primum finitce sunt magnitudinis, desinunt 
esse momenta. Finiri enim repugnat aliquatenus perpetuo 
eorum incremento vel decremento. Intelligenda sunt principia 
jamjam nascentia finitorum magnitudinum." In the second 
edition the two sentences which we print in italics are replaced 
by the following: "Particulse finitae non sunt momenta sed 
quantitates ipsse ex momentis genitae." Through the difficulty 
of the phrases in both extracts, this much distinctly appears, 
that in the first, moments are infinitely small quantities. What 


else they are in the second is not clear. [35] In the Quadrature 
of Curves of 1704, the infinitely small quantity is completely 
abandoned. It has been shown that in the Method of Fluxions 
Newton rejected terms involving the quantity 0, because 
they are infinitely small compared with other terms. This 
reasoning is evidently erroneous; for as long as is a quantity 
though ever so small, this rejection cannot be made without 
affecting the result. Newton seems to have felt this, for in 
the Quadrature of Curves he remarked that "in mathematics 
the minutest errors are not to be neglected" (errores quam 
minimi in rebus mathematicis non sunt contemnendi). 

The early distinction between the system of Newton and 
Leibniz lies in this, that Newton, holding to the conception 
of velocity or fluxion, used the infinitely small increment as a 
means of determining it, while with Leibniz the relation of the 
infinitely small increments is itself the object of determination. 
The difference between the two rests mainly upon a difference 
in the mode of generating quantities. [35] 

We give Newton's statement of the method of fluxions 
or rates, as given in the introduction to his Quadrature of 
Curves. "I consider mathematical quantities in this place 
not as consisting of very small parts, but as described by a 
continued motion. Lines are described, and thereby generated, 
not by the apposition of parts, but by the continued motion of 
points; superficies by the motion of lines; solids by the motion 
of superficies; angles by the rotation of the sides; portions of 
time by continual flux: and so on in other quantities. These 
geneses really take place in the nature of things, and are daily 



seen in the motion of bodies. . . . 

"Fluxions are, as near as we please (quam proxime), as the 
increments of fluents generated in times, equal and as small as 
possible, and to speak accurately, they are in the prime ratio 
of nascent increments; yet they can be expressed by any lines 
whatever, which are proportional to them." 

Newton exemplifies this last assertion by the problem of 
tangency: Let AB be the abscissa, BC the ordinate, VCH the 
tangent, Ec the increment of the ordinate, which produced 
meets VH at T, and Cc the increment of the curve. The right 
line Cc being produced to K, there are formed three small 
triangles, the rectilinear CEc, the mixtilinear CEc, and the 
rectilinear GET . Of these, the first is evidently the smallest, 
and the last the greatest. Now suppose the ordinate be to move 
into the place BC, so that the point c exactly coincides with 

the point C; CK, 
and therefore the 
curve Cc, is coin- 
cident with the tan- 
gent CH, Ecis abso- 
lutely equal to ET, 
and the mixtilinear 
evanescent triangle 
CEc is, in the last form, similar to the triangle CET; and its 
evanescent sides CE, Ec, Cc, will be proportional to CE, ET, 
and CT, the sides of the triangle CET. Hence it follows that 
the fluxions of the lines AB, BC, AC, being in the last ratio 
of their evanescent increments, are proportional to the sides 


of the triangle CET, or, which is all one, of the triangle VBC 
similar thereunto. As long as the points C and c are distant 
from each other by an interval, however small, the line CK 
will stand apart by a small angle from the tangent CH. But 
when CK coincides with CH, and the lines CE, Ec, cC reach 
their ultimate ratios, then the points C and c accurately 
coincide and are one and the same. Newton then adds that 
"in mathematics the minutest errors are not to be neglected." 
This is plainly a rejection of the postulates of Leibniz. The 
doctrine of infinitely small quantities is here renounced in a 
manner which would lead one to suppose that Newton had 
never held it himself. Thus it appears that Newton's doctrine 
was different in different periods. Though, in the above 
reasoning, the Charybdis of infinitesimals is safely avoided, 
the dangers of a Scylla stare us in the face. We are required 
to believe that a point may be considered a triangle, or that a 
triangle can be inscribed in a point; nay, that three dissimilar 
triangles become similar and equal when they have reached 
their ultimate form in one and the same point. 

In the introduction to the Quadrature of Curves the fluxion 
of x n is determined as follows:— 

"In the same time that x, by flowing, becomes x + 0, the 
power x n becomes (x + 0) n , i. e. by the method of infinite series 

x n + nOx 71 ' 1 + ^-^oV" 2 + etc., 

and the increments 


and n0x n_i + ' - '" ^x n ~ l + etc., 

n-l , n - n n 2 n-2 


are to one another as 


1 to nx n ~ l + — - — Ox 11 ' 2 + etc. 

"Let now the increments vanish, and their last proportion 
will be 1 to nx n_1 : hence the fluxion of the quantity x is to 
the fluxion of the quantity x n as 1 : nx n_1 . 

"The fluxion of lines, straight or curved, in all cases 
whatever, as also the fluxions of superficies, angles, and other 
quantities, can be obtained in the same manner by the method 
of prime and ultimate ratios. But to establish in this way the 
analysis of infinite quantities, and to investigate prime and 
ultimate ratios of finite quantities, nascent or evanescent, is 
in harmony with the geometry of the ancients; and I have 
endeavoured to show that, in the method of fluxions, it 
is not necessary to introduce into geometry infinitely small 
quantities." This mode of differentiating does not remove all 
the difficulties connected with the subject. When becomes 
nothing, then we get the ratio - = ra"" 1 , which needs further 
elucidation. Indeed, the method of Newton, as delivered 
by himself, is encumbered with difficulties and objections. 
Among the ablest admirers of Newton, there have been 
obstinate disputes respecting his explanation of his method 
of "prime and ultimate ratios." 

The so-called "method of limits" is frequently attributed 
to Newton, but the pure method of limits was never adopted 
by him as his method of constructing the calculus. All he did 
was to establish in his Principia certain principles which are 
applicable to that method, but which he used for a different 


purpose. The first lemma of the first book has been made the 
foundation of the method of limits:— 

"Quantities and the ratios of quantities, which in any finite 
time converge continually to equality, and before the end of 
that time approach nearer the one to the other than by any 
given difference, become ultimately equal." 

In this, as well as in the lemmas following this, there are 
obscurities and difficulties. Newton appears to teach that a 
variable quantity and its limit will ultimately coincide and 
be equal. But it is now generally agreed that in the clearest 
statements which have been made of the theory of limits, the 
variable does not actually reach its limit, though the variable 
may approach it as near as we please. 

The full title of Newton's Principia is Philosophic Naturalis 
Principia Mathematica. It was printed in 1687 under the 
direction, and at the expense, of Dr. Edmund Halley. A 
second edition was brought out in 1713 with many alterations 
and improvements, and accompanied by a preface from 
Mr. Cotes. It was sold out in a few months, but a pirated 
edition published in Amsterdam supplied the demand. [34] 
The third and last edition which appeared in England during 
Newton's lifetime was published in 1726 by Henry Pemberton. 
The Principia consists of three books, of which the first 
two, constituting the great bulk of the work, treat of the 
mathematical principles of natural philosophy, namely, the 
laws and conditions of motions and forces. In the third book 
is drawn up the constitution of the universe as deduced from 
the foregoing principles. The great principle underlying this 


memorable work is that of universal gravitation. The first 
book was completed on April 28, 1686. After the remarkably 
short period of three months, the second book was finished. 
The third book is the result of the next nine or ten months' 
labours. It is only a sketch of a much more extended 
elaboration of the subject which he had planned, but which 
was never brought to completion. 

The law of gravitation is enunciated in the first book. Its 
discovery envelops the name of Newton in a halo of perpetual 
glory. The current version of the discovery is as follows: it 
was conjectured by Hooke, Huygens, Halley, Wren, Newton, 
and others, that, if Kepler's third law was true (its absolute 
accuracy was doubted at that time), then the attraction 
between the earth and other members of the solar system 
varied inversely as the square of the distance. But the proof of 
the truth or falsity of the guess was wanting. In 1666 Newton 
reasoned, in substance, that if g represent the acceleration of 
gravity on the surface of the earth, r be the earth's radius, 
R the distance of the moon from the earth, T the time of lunar 
revolution, and a a degree at the equator, then, if the law is 

r 2 o R ^ (R\ 3 

^ = 47r V' or5= TH7j ' 180a - 

The data at Newton's command gave R = 60 Ar , T = 2, 360, 628 
seconds, but a only 60 instead of 69^ English miles. This 
wrong value of a rendered the calculated value of g smaller 
than its true value, as known from actual measurement. It 
looked as though the law of inverse squares were not the true 


law, and Newton laid the calculation aside. In 1684 he casually 
ascertained at a meeting of the Royal Society that Jean Picard 
had measured an arc of the meridian, and obtained a more 
accurate value for the earth's radius. Taking the corrected 
value for a, he found a figure for g which corresponded to the 
known value. Thus the law of inverse squares was verified. 
In a scholium in the Principia, Newton acknowledged his 
indebtedness to Huygens for the laws on centrifugal force 
employed in his calculation. 

The perusal by the astronomer Adams of a great mass 
of unpublished letters and manuscripts of Newton forming 
the Portsmouth collection (which remained private property 
until 1872, when its owner placed it in the hands of the 
University of Cambridge) seems to indicate that the difficulties 
encountered by Newton in the above calculation were of a 
different nature. According to Adams, Newton's numerical 
verification was fairly complete in 1666, but Newton had not 
been able to determine what the attraction of a spherical shell 
upon an external point would be. His letters to Halley show 
that he did not suppose the earth to attract as though all 
its mass were concentrated into a point at the centre. He 
could not have asserted, therefore, that the assumed law of 
gravity was verified by the figures, though for long distances 
he might have claimed that it yielded close approximations. 
When Halley visited Newton in 1684, he requested Newton to 
determine what the orbit of a planet would be if the law of 
attraction were that of inverse squares. Newton had solved a 
similar problem for Hooke in 1679, and replied at once that it 


was an ellipse. After Halley's visit, Newton, with Picard's new 
value for the earth's radius, reviewed his early calculation, 
and was able to show that if the distances between the bodies 
in the solar system were so great that the bodies might be 
considered as points, then their motions were in accordance 
with the assumed law of gravitation. In 1685 he completed 
his discovery by showing that a sphere whose density at any 
point depends only on the distance from the centre attracts 
an external point as though its whole mass were concentrated 
at the centre. [34] 

Newton's unpublished manuscripts in the Portsmouth col- 
lection show that he had worked out, by means of fluxions and 
fluents, his lunar calculations to a higher degree of approxima- 
tion than that given in the Principia, but that he was unable 
to interpret his results geometrically. The papers in that col- 
lection throw light upon the mode by which Newton arrived 
at some of the results in the Principia, as, for instance, the 
famous construction in Book II., Prop. 25, which is unproved 
in the Principia, but is demonstrated by him twice in a draft 
of a letter to David Gregory, of Oxford. [34] 

It is chiefly upon the Principia that the fame of Newton 
rests. Brewster calls it "the brightest page in the records of 
human reason." Let us listen, for a moment, to the comments 
of Laplace, the foremost among those followers of Newton 
who grappled with the subtle problems of the motions of 
planets under the influence of gravitation: "Newton has well 
established the existence of the principle which he had the 
merit of discovering, but the development of its consequences 


and advantages has been the work of the successors of this 
great mathematician. The imperfection of the infinitesimal 
calculus, when first discovered, did not allow him completely 
to resolve the difficult problems which the theory of the 
universe offers; and he was oftentimes forced to give mere 
hints, which were always uncertain till confirmed by rigorous 
analysis. Notwithstanding these unavoidable defects, the 
importance and the generality of his discoveries respecting 
the system of the universe, and the most interesting points 
of natural philosophy, the great number of profound and 
original views, which have been the origin of the most brilliant 
discoveries of the mathematicians of the last century, which 
were all presented with much elegance, will insure to the 
Principia a lasting pre-eminence over all other productions of 
the human mind." 

Newton's Arithmetica Universalis, consisting of algebraical 
lectures delivered by him during the first nine years he was 
professor at Cambridge, were published in 1707, or more 
than thirty years after they were written. This work was 
published by Mr. Whiston. We are not accurately informed 
how Mr. Whiston came in possession of it, but according to 
some authorities its publication was a breach of confidence on 
his part. 

The Arithmetica Universalis contains new and important 
results on the theory of equations. His theorem on the sums 
of powers of roots is well known. Newton showed that in 
equations with real coefficients, imaginary roots always occur 
in pairs. His inventive genius is grandly displayed in his rule 


for determining the inferior limit of the number of imaginary 
roots, and the superior limits for the number of positive and 
negative roots. Though less expeditious than Descartes', 
Newton's rule always gives as close, and generally closer, 
limits to the number of positive and negative roots. Newton 
did not prove his rule. It awaited demonstration for a century 
and a half, until, at last, Sylvester established a remarkable 
general theorem which includes Newton's rule as a special 

The treatise on Method of Fluxions contains Newton's 
method of approximating to the roots of numerical equations. 
This is simply the method of Vieta improved. The same 
treatise contains "Newton's parallelogram," which enabled 
him, in an equation, f(x, y) = 0, to find a series in powers of x 
equal to the variable y. The great utility of this rule lay in 
its determining the form of the series; for, as soon as the law 
was known by which the exponents in the series vary, then the 
expansion could be effected by the method of indeterminate 
coefficients. The rule is still used in determining the infinite 
branches to curves, or their figure at multiple points. Newton 
gave no proof for it, nor any clue as to how he discovered it. 
The proof was supplied half a century later, by Kaestner and 
Cramer, independently. [37] 

In 1704 was published, as an appendix to the Opticks, 
the Enumeratio linearum tertii ordinis, which contains the- 
orems on the theory of curves. Newton divides cubics into 
seventy-two species, arranged in larger groups, for which 
his commentators have supplied the names "genera" and 


"classes," recognising fourteen of the former and seven (or 
four) of the latter. He overlooked six species demanded by his 
principles of classification, and afterwards added by Stirling, 
Murdoch, and Cramer. He enunciates the remarkable theorem 
that the five species which he names "divergent parabolas" 
give by their projection every cubic curve whatever. As a rule, 
the tract contains no proofs. It has been the subject of frequent 
conjecture how Newton deduced his results. Recently we have 
gotten at the facts, since much of the analysis used by Newton 
and a few additional theorems have been discovered among 
the Portsmouth papers. An account of the four holograph 
manuscripts on this subject has been published by W. W. 
Rouse Ball, in the Transactions of the London Mathematical 
Society (vol. xx., pp. 104-143). It is interesting to observe 
how Newton begins his research on the classification of cubic 
curves by the algebraic method, but, finding it laborious, 
attacks the problem geometrically, and afterwards returns 
again to analysis. [36] 

Space does not permit us to do more than merely men- 
tion Newton's prolonged researches in other departments of 
science. He conducted a long series of experiments in optics 
and is the author of the corpuscular theory of light. The 
last of a number of papers on optics, which he contributed to 
the Royal Society, 1687, elaborates the theory of "fits." He 
explained the decomposition of light and the theory of the 
rainbow. By him were invented the reflecting telescope and 
the sextant (afterwards re-discovered by Thomas Godfrey of 
Philadelphia [2] and by John Hadley). He deduced a theo- 


retical expression for the velocity of sound in air, engaged in 
experiments on chemistry, elasticity, magnetism, and the law 
of cooling, and entered upon geological speculations. 

During the two years following the close of 1692, Newton 
suffered from insomnia and nervous irritability. Some thought 
that he laboured under temporary mental aberration. Though 
he recovered his tranquillity and strength of mind, the 
time of great discoveries was over; he would study out 
questions propounded to him, but no longer did he by his 
own accord enter upon new fields of research. The most 
noted investigation after his sickness was the testing of his 
lunar theory by the observations of Flamsteed, the astronomer 
royal. In 1695 he was appointed warden, and in 1699 master, 
of the mint, which office he held until his death. His body was 
interred in Westminster Abbey, where in 1731 a magnificent 
monument was erected, bearing an inscription ending with, 
"Sibi gratulentur mortales tale tantumque exstitisse humani 
generis decus." It is not true that the Binomial Theorem is 
also engraved on it. 

We pass to Leibniz, the second and independent inventor 
of the calculus. Gottfried Wilhelm Leibniz (1646-1716) 
was born in Leipzig. No period in the history of any 
civilised nation could have been less favourable for literary 
and scientific pursuits than the middle of the seventeenth 
century in Germany. Yet circumstances seem to have happily 
combined to bestow on the youthful genius an education hardly 
otherwise obtainable during this darkest period of German 
history. He was brought early in contact with the best of 


the culture then existing. In his fifteenth year he entered the 
University of Leipzig. Though law was his principal study, 
he applied himself with great diligence to every branch of 
knowledge. Instruction in German universities was then very 
low. The higher mathematics was not taught at all. We are 
told that a certain John Kuhn lectured on Euclid's Elements, 
but that his lectures were so obscure that none except Leibniz 
could understand them. Later on, Leibniz attended, for a half- 
year, at Jena, the lectures of Erhard Weigel, a philosopher 
and mathematician of local reputation. In 1666 Leibniz 
published a treatise, De Arte Combinatoria, in which he does 
not pass beyond the rudiments of mathematics. Other theses 
written by him at this time were metaphysical and juristical 
in character. A fortunate circumstance led Leibniz abroad. 
In 1672 he was sent by Baron Boineburg on a political mission 
to Paris. He there formed the acquaintance of the most 
distinguished men of the age. Among these was Huygens, 
who presented a copy of his work on the oscillation of the 
pendulum to Leibniz, and first led the gifted young German 
to the study of higher mathematics. In 1673 Leibniz went 
to London, and remained there from January till March. He 
there became incidentally acquainted with the mathematician 
Pell, to whom he explained a method he had found on the 
summation of series of numbers by their differences. Pell told 
him that a similar formula had been published by Mouton as 
early as 1670, and then called his attention to Mercator's work 
on the rectification of the parabola. While in London, Leibniz 
exhibited to the Royal Society his arithmetical machine, which 


was similar to Pascal's, but more efficient and perfect. After 
his return to Paris, he had the leisure to study mathematics 
more systematically. With indomitable energy he set about 
removing his ignorance of higher mathematics. Huygens 
was his principal master. He studied the geometric works of 
Descartes, Honorarius Fabri, Gregory St. Vincent, and Pascal. 
A careful study of infinite series led him to the discovery of 
the following expression for the ratio of the circumference to 
the diameter of the circle, previously discovered by James 

This elegant series was found in the same way as Mercator's 
on the hyperbola. Huygens was highly pleased with it and 
urged him on to new investigations. Leibniz entered into 
a detailed study of the quadrature of curves and thereby 
became intimately acquainted with the higher mathematics. 
Among the papers of Leibniz is still found a manuscript on 
quadratures, written before he left Paris in 1676, but which 
was never printed by him. The more important parts of it were 
embodied in articles published later in the Acta Eruditorum. 
In the study of Cartesian geometry the attention of Leibniz 
was drawn early to the direct and inverse problems of 
tangents. The direct problem had been solved by Descartes 
for the simplest curves only; while the inverse had completely 
transcended the power of his analysis. Leibniz investigated 
both problems for any curve; he constructed what he called 
the triangulum characteristicum — an infinitely small triangle 
between the infinitely small part of the curve coinciding 


with the tangent, and the differences of the ordinates and 
abscissas. A curve is here considered to be a polygon. The 
triangulum characteristicum is similar to the triangle formed 
by the tangent, the ordinate of the point of contact, and 
the sub-tangent, as well as to that between the ordinate, 
normal, and sub-normal. It was first employed by Barrow in 
England, but appears to have been re-invented by Leibniz. 
From it Leibniz observed the connection existing between the 
direct and inverse problems of tangents. He saw also that the 
latter could be carried back to the quadrature of curves. All 
these results are contained in a manuscript of Leibniz, written 
in 1673. One mode used by him in effecting quadratures was 
as follows: The rectangle formed by a sub-tangent p and an 
element a (i.e. infinitely small part of the abscissa) is equal 
to the rectangle formed by the ordinate y and the element I 
of that ordinate; or in symbols, pa = yl. But the summation 
of these rectangles from zero on gives a right triangle equal 
to half the square of the ordinate. Thus, using Cavalieri's 
notation, he gets 

y 2 

omn. pa = omn. yl = — (omn. meaning omnia, all). 
But y = omn. I; hence 

omn. omn. I 

I omn. / 2 

a 2a 

This equation is especially interesting, since it is here that 
Leibniz first introduces a new notation. He says: "It will be 
useful to write J for omn., as / 1 for omn. I, that is, the sum of 


the f s" ; he then writes the equation thus:— 

2a J J a 
From this he deduced the simplest integrals, such as 

/x = y, (x + y) = / x + j y. 

Since the symbol of summation f raises the dimensions, he 

concluded that the opposite calculus, or that of differences d, 

would lower them. Thus, if J I = ya, then I = —. The 

symbol d was at first placed by Leibniz in the denominator, 

because the lowering of the power of a term was brought about 

in ordinary calculation by division. The manuscript giving 

the above is dated October 29th, 1675. [39] This, then, was 

the memorable day on which the notation of the new calculus 

came to be, — a notation which contributed enormously to the 

rapid growth and perfect development of the calculus. 

Leibniz proceeded to apply his new calculus to the solution 

of certain problems then grouped together under the name 

of the Inverse Problems of Tangents. He found the cubical 

parabola to be the solution to the following: To find the curve 

in which the sub-normal is reciprocally proportional to the 

ordinate. The correctness of his solution was tested by him by 

applying to the result Sluze's method of tangents and reasoning 

backwards to the original supposition. In the solution of the 


third problem he changes his notation from - to the now usual 

notation dx. It is worthy of remark that in these investigations, 

Leibniz nowhere explains the significance of dx and dy, except 



at one place in a marginal note: "Idem est dx et -, id est, 


differentia inter duas x proximas." Nor does he use the term 
differential, but always difference. Not till ten years later, in 
the Acta Eruditorum, did he give further explanations of these 
symbols. What he aimed at principally was to determine the 
change an expression undergoes when the symbol / or d is 
placed before it. It may be a consolation to students wrestling 
with the elements of the differential calculus to know that it 

required Leibniz considerable thought and attention [39] to 

determine whether dx dy is the same as dixy), and — the same 

x d y 

as d—. After considering these questions at the close of one of 


his manuscripts, he concluded that the expressions were not 
the same, though he could not give the true value for each. 
Ten days later, in a manuscript dated November 21, 1675, he 
found the equation ydx = dxy — x dy, giving an expression 
for d(xy), which he observed to be true for all curves. He 
succeeded also in eliminating dx from a differential equation, 
so that it contained only dy, and thereby led to the solution 
of the problem under consideration. "Behold, a most elegant 
way by which the problems of the inverse methods of tangents 
are solved, or at least are reduced to quadratures!" Thus 
he saw clearly that the inverse problems of tangents could 
be solved by quadratures, or, in other words, by the integral 
calculus. In course of a half-year he discovered that the direct 
problem of tangents, too, yielded to the power of his new 
calculus, and that thereby a more general solution than that 
of Descartes could be obtained. He succeeded in solving all the 
special problems of this kind, which had been left unsolved by 


Descartes. Of these we mention only the celebrated problem 
proposed to Descartes by De Beaune, viz. to find the curve 
whose ordinate is to its sub-tangent as a given line is to that 
part of the ordinate which lies between the curve and a line 
drawn from the vertex of the curve at a given inclination to 
the axis. 

Such was, in brief, the progress in the evolution of the new 
calculus made by Leibniz during his stay in Paris. Before his 
departure, in October, 1676, he found himself in possession 
of the most elementary rules and formulae of the infinitesimal 

From Paris, Leibniz returned to Hanover by way of London 
and Amsterdam. In London he met Collins, who showed him 
a part of his scientific correspondence. Of this we shall speak 
later. In Amsterdam he discussed mathematics with Sluze, 
and became satisfied that his own method of constructing 
tangents not only accomplished all that Sluze's did, but 
even more, since it could be extended to three variables, 
by which tangent planes to surfaces could be found; and 
especially, since neither irrationals nor fractions prevented 
the immediate application of his method. 

In a paper of July 11, 1677, Leibniz gave correct rules for 

the differentiation of sums, products, quotients, powers, and 

roots. He had given the differentials of a few negative and 

fractional powers, as early as November, 1676, but had made 

some mistakes. For d^/x he had given the erroneous value —=, 

, _i 1 V x 

and in another place the value — Ax 2 ; for d^- occurs in one 

2 x A 

place the wrong value, — k, while a few lines lower is given 


3 . 

— j, its correct value. 

In 1682 was founded in Berlin the Acta Eruditorum, a 
journal usually known by the name of Leipzig Acts. It was a 
partial imitation of the French Journal des Savans (founded 
in 1665), and the literary and scientific review published in 
Germany Leibniz was a frequent contributor. Tschirnhaus, 
who had studied mathematics in Paris with Leibniz, and who 
was familiar with the new analysis of Leibniz, published in 
the Acta Eruditorum a paper on quadratures, which consists 
principally of subject-matter communicated by Leibniz to 
Tschirnhaus during a controversy which they had had on this 
subject. Fearing that Tschirnhaus might claim as his own 
and publish the notation and rules of the differential calculus, 
Leibniz decided, at last, to make public the fruits of his 
inventions. In 1684, or nine years after the new calculus first 
dawned upon the mind of Leibniz, and nineteen years after 
Newton first worked at fluxions, and three years before the 
publication of Newton's Principia, Leibniz published, in the 
Leipzig Acts, his first paper on the differential calculus. He 
was unwilling to give to the world all his treasures, but chose 
those parts of his work which were most abstruse and least 
perspicuous. This epoch-making paper of only six pages bears 
the title: "Nova methodus pro maximis et minimis, itemque 
tangentibus, quae nee fractas nee irrationales quantitates 
moratur, et singulare pro illis calculi genus." The rules of 
calculation are briefly stated without proof, and the meaning 
of dx and dy is not made clear. It has been inferred from this 
that Leibniz himself had no definite and settled ideas on this 


subject. Are dy and dx finite or infinitesimal quantities? At 
first they appear, indeed, to have been taken as finite, when 
he says: "We now call any line selected at random dx, then 
we designate the line which is to dx as y is to the sub-tangent, 
by dy, which is the difference of y." Leibniz then ascertains, 
by his calculus, in what way a ray of light passing through 
two differently refracting media, can travel easiest from one 
point to another; and then closes his article by giving his 
solution, in a few words, of De Beaune's problem. Two 
years later (1686) Leibniz published in the Acta Eruditorum a 
paper containing the rudiments of the integral calculus. The 
quantities dx and dy are there treated as infinitely small. He 
showed that by the use of his notation, the properties of curves 
could be fully expressed by equations. Thus the equation 



\/2x — x 1 + / 

V2x — x 2 
characterises the cycloid. [38] 

The great invention of Leibniz, now made public by his 
articles in the Leipzig Acts, made little impression upon the 
mass of mathematicians. In Germany no one comprehended 
the new calculus except Tschirnhaus, who remained indifferent 
to it. The author's statements were too short and succinct 
to make the calculus generally understood. The first to 
recognise its importance and to take up the study of it 
were two foreigners, — the Scotchman John Craig, and the 
Swiss James Bernoulli. The latter wrote Leibniz a letter 
in 1687, wishing to be initiated into the mysteries of the 
new analysis. Leibniz was then travelling abroad, so that 


this letter remained unanswered till 1690. James Bernoulli 
succeeded, meanwhile, by close application, in uncovering 
the secrets of the differential calculus without assistance. 
He and his brother John proved to be mathematicians of 
exceptional power. They applied themselves to the new 
science with a success and to an extent which made Leibniz 
declare that it was as much theirs as his. Leibniz carried on 
an extensive correspondence with them, as well as with other 
mathematicians. In a letter to John Bernoulli he suggests, 
among other things, that the integral calculus be improved 
by reducing integrals back to certain fundamental irreducible 
forms. The integration of logarithmic expressions was then 
studied. The writings of Leibniz contain many innovations, 
and anticipations of since prominent methods. Thus he made 
use of variable parameters, laid the foundation of analysis 
in situ, introduced the first notion of determinants in his 
effort to simplify the expression arising in the elimination of 
the unknown quantities from a set of linear equations. He 
resorted to the device of breaking up certain fractions into the 
sum of other fractions for the purpose of easier integration; he 
explicitly assumed the principle of continuity; he gave the first 
instance of a "singular solution," and laid the foundation to 
the theory of envelopes in two papers, one of which contains for 
the first time the terms co-ordinate and axes of co-ordinates. 
He wrote on osculating curves, but his paper contained the 
error (pointed out by John Bernoulli, but not admitted by 
him) that an osculating circle will necessarily cut a curve in 
four consecutive points. Well known is his theorem on the 


rath differential coefficient of the product of two functions 
of a variable. Of his many papers on mechanics, some are 
valuable, while others contain grave errors. 

Before tracing the further development of the calculus we 
shall sketch the history of that long and bitter controversy 
between English and Continental mathematicians on the 
invention of the calculus. The question was, did Leibniz 
invent it independently of Newton, or was he a plagiarist? 

We must begin with the early correspondence between the 
parties appearing in this dispute. Newton had begun using his 
notation of fluxions in 1666. [41] In 1669 Barrow sent Collins 
Newton's tract, De Analysi per Equationes, etc. 

The first visit of Leibniz to London extended from the 
11th of January until March, 1673. He was in the habit of 
committing to writing important scientific communications 
received from others. In 1890 Gerhardt discovered in the 
royal library at Hanover a sheet of manuscript with notes 
taken by Leibniz during this journey. [40] They are headed 
"Observata Philosophica in itinere Anglicano sub initium 
anni 1673." The sheet is divided by horizontal lines into sec- 
tions. The sections given to Chymica, Mechanica, Magnetica, 
Botanica, Anatomica, Medica, Miscellanea, contain extensive 
memoranda, while those devoted to mathematics have very 
few notes. Under Geometrica he says only this: "Tangentes 
omnium figurarum. Figurarum geometricarum explicatio per 
motum puncti in moto lati." We suspect from this that 
Leibniz had read Barrow's lectures. Newton is referred to only 
under Optica. Evidently Leibniz did not obtain a knowledge 


of fluxions during this visit to London, nor is it claimed that 
he did by his opponents. 

Various letters of Newton, Collins, and others, up to the 
beginning of 1676, state that Newton invented a method by 
which tangents could be drawn without the necessity of freeing 
their equations from irrational terms. Leibniz announced in 
1674 to Oldenburg, then secretary of the Royal Society, that 
he possessed very general analytical methods, by which he 
had found theorems of great importance on the quadrature 
of the circle by means of series. In answer, Oldenburg stated 
Newton and James Gregory had also discovered methods of 
quadratures, which extended to the circle. Leibniz desired 
to have these methods communicated to him; and Newton, 
at the request of Oldenburg and Collins, wrote to the former 
the celebrated letters of June 13 and October 24, 1676. 
The first contained the Binomial Theorem and a variety of 
other matters relating to infinite series and quadratures; but 
nothing directly on the method of fluxions. Leibniz in reply 
speaks in the highest terms of what Newton had done, and 
requests further explanation. Newton in his second letter just 
mentioned explains the way in which he found the Binomial 
Theorem, and also communicates his method of fluxions and 
fluents in form of an anagram in which all the letters in the 
sentence communicated were placed in alphabetical order. 
Thus Newton says that his method of drawing tangents was 

6 ace da? 13 eff 7 i 3/ 9n Ao Aqrr As 9t Ylvx. 

The sentence was, "Data aequatione quotcunque fluentes 


quantitates involvente fluxiones invenire, et vice versa." 
( "Having any given equation involving never so many flowing 
quantities, to find the fluxions, and vice versa." ) Surely this 
anagram afforded no hint. Leibniz wrote a reply to Collins, 
in which, without any desire of concealment, he explained the 
principle, notation, and the use of the differential calculus. 

The death of Oldenburg brought this correspondence to a 
close. Nothing material happened till 1684, when Leibniz 
published his first paper on the differential calculus in the 
Leipzig Acts, so that while Newton's claim to the priority of 
invention must be admitted by all, it must also be granted that 
Leibniz was the first to give the full benefit of the calculus to 
the world. Thus, while Newton's invention remained a secret, 
communicated only to a few friends, the calculus of Leibniz was 
spreading over the Continent. No rivalry or hostility existed, 
as yet, between the illustrious scientists. Newton expressed a 
very favourable opinion of Leibniz's inventions, known to him 
through the above correspondence with Oldenburg, in the 
following celebrated scholium (Principia, first edition, 1687, 
Book II., Prop. 7, scholium):— 

"In letters which went between me and that most excellent 
geometer, G. G. Leibniz, ten years ago, when I signified that 
I was in the knowledge of a method of determining maxima 
and minima, of drawing tangents, and the like, and when 
I concealed it in transposed letters involving this sentence 
(Data aequatione, etc., above cited), that most distinguished 
man wrote back that he had also fallen upon a method of 
the same kind, and communicated his method, which hardly 


differed from mine, except in his forms of words and symbols." 
As regards this passage, we shall see that Newton was 
afterwards weak enough, as De Morgan says: "First, to 
deny the plain and obvious meaning, and secondly, to omit 
it entirely from the third edition of the Principia." On 
the Continent, great progress was made in the calculus by 
Leibniz and his coadjutors, the brothers James and John 
Bernoulli, and Marquis de l'Hospital. In 1695 Wallis informed 
Newton by letter that "he had heard that his notions of 
fluxions passed in Holland with great applause by the name 
of 'Leibniz's Calculus Differentialis.''' Accordingly Wallis 
stated in the preface to a volume of his works that the calculus 
differentialis was Newton's method of fluxions which had been 
communicated to Leibniz in the Oldenburg letters. A review 
of Wallis' works, in the Leipzig Acts for 1696, reminded the 
reader of Newton's own admission in the scholium above cited. 
For fifteen years Leibniz had enjoyed unchallenged the 
honour of being the inventor of his calculus. But in 1699 
Fato de Duillier, a Swiss, who had settled in England, stated 
in a mathematical paper, presented to the Royal Society, 
his conviction that Newton was the first inventor; adding 
that, whether Leibniz, the second inventor, had borrowed 
anything from the other, he would leave to the judgment of 
those who had seen the letters and manuscripts of Newton. 
This was the first distinct insinuation of plagiarism. It would 
seem that the English mathematicians had for some time 
been cherishing suspicions unfavourable to Leibniz. A feeling 
had doubtless long prevailed that Leibniz, during his second 


visit to London in 1676, had or might have seen among 
the papers of Collins, Newton's Analysis per dsquationes, etc., 
which contained applications of the fluxionary method, but no 
systematic development or explanation of it. Leibniz certainly 
did see at least part of this tract. During the week spent in 
London, he took note of whatever interested him among the 
letters and papers of Collins. His memoranda discovered by 
Gerhardt in 1849 in the Hanover library fill two sheets. [40] 
The one bearing on our question is headed "Excerpta ex 
tractatu Newtoni Msc. de Analysi per aequationes numero 
terminorum infinitas." The notes are very brief, excepting 
those De Resolutione (Bquationum affectarum, of which there 
is an almost complete copy. This part was evidently new 
to him. If he examined Newton's entire tract, the other 
parts did not particularly impress him. From it he seems to 
have gained nothing pertaining to the infinitesimal calculus. 
By the previous introduction of his own algorithm he had 
made greater progress than by what came to his knowledge in 
London. Nothing mathematical that he had received engaged 
his thoughts in the immediate future, for on his way back 
to Holland he composed a lengthy dialogue on mechanical 

Duillier's insinuations lighted up a flame of discord which a 
whole century was hardly sufficient to extinguish. Leibniz, who 
had never contested the priority of Newton's discovery, and 
who appeared to be quite satisfied with Newton's admission in 
his scholium, now appears for the first time in the controversy. 
He made an animated reply in the Leipzig Acts, and complained 


to the Royal Society of the injustice done him. 

Here the affair rested for some time. In the Quadrature of 
Curves, published 1704, for the first time, a formal exposition 
of the method and notation of fluxions was made public. 
In 1705 appeared an unfavourable review of this in the Leipzig 
Acts, stating that Newton uses and always has used fluxions 
for the differences of Leibniz. This was considered by Newton's 
friends an imputation of plagiarism on the part of their chief, 
but this interpretation was always strenuously resisted by 
Leibniz. Keill, professor of astronomy at Oxford, undertook 
with more zeal than judgment the defence of Newton. In 
a paper inserted in the Philosophical Transactions of 1708, 
he claimed that Newton was the first inventor of fluxions 
and "that the same calculus was afterward published by 
Leibniz, the name and the mode of notation being changed." 
Leibniz complained to the secretary of the Royal Society of 
bad treatment and requested the interference of that body to 
induce Keill to disavow the intention of imputing fraud. Keill 
was not made to retract his accusation; on the contrary, was 
authorised by Newton and the Royal Society to explain and 
defend his statement. This he did in a long letter. Leibniz 
thereupon complained that the charge was now more open 
than before, and appealed for justice to the Royal Society and 
to Newton himself. The Royal Society, thus appealed to as a 
judge, appointed a committee which collected and reported 
upon a large mass of documents — mostly letters from and 
to Newton, Leibniz, Wallis, Collins, etc. This report, called 
the Commercium Epistolicum, appeared in the year 1712 and 


again in 1725, with a Recensio prefixed, and additional notes 
byKeill. The final conclusion in the Commercium Epistolicum 
was that Newton was the first inventor. But this was not to 
the point. The question was not whether Newton was the first 
inventor, but whether Leibniz had stolen the method. The 
committee had not formally ventured to assert their belief 
that Leibniz was a plagiarist. Yet there runs throughout 
the document a desire of proving Leibniz guilty of more than 
they meant positively to affirm. Leibniz protested only in 
private letters against the proceeding of the Royal Society, 
declaring that he would not answer an argument so weak. 
John Bernoulli, in a letter to Leibniz, which was published 
later in an anonymous tract, is as decidedly unfair towards 
Newton as the friends of the latter had been towards Leibniz. 
Keill replied, and then Newton and Leibniz appear as mutual 
accusers in several letters addressed to third parties. In a 
letter to Conti, April 9, 1716, Leibniz again reminded Newton 
of the admission he had made in the scholium, which he was 
now desirous of disavowing; Leibniz also states that he always 
believed Newton, but that, seeing him connive at accusations 
which he must have known to be false, it was natural that 
he (Leibniz) should begin to doubt. Newton did not reply 
to this letter, but circulated some remarks among his friends 
which he published immediately after hearing of the death of 
Leibniz, November 14, 1716. This paper of Newton gives the 
following explanation pertaining to the scholium in question: 
"He [Leibniz] pretends that in my book of principles I allowed 
him the invention of the calculus differentialis, independently 


of my own; and that to attribute this invention to myself is 
contrary to my knowledge there avowed. But in the paragraph 
there referred unto I do not find one word to this purpose." In 
the third edition of the Principia, 1726, Newton omitted the 
scholium and substituted in its place another, in which the 
name of Leibniz does not appear. 

National pride and party feeling long prevented the adoption 
of impartial opinions in England, but now it is generally 
admitted by nearly all familiar with the matter, that Leibniz 
really was an independent inventor. Perhaps the most telling 
evidence to show that Leibniz was an independent inventor is 
found in the study of his mathematical papers (collected and 
edited by C. I. Gerhardt, in six volumes, Berlin, 1849-1860), 
which point out a gradual and natural evolution of the rules 
of the calculus in his own mind. "There was throughout the 
whole dispute," says De Morgan, "a confusion between the 
knowledge of fluxions or differentials and that of a calculus 
of fluxions or differentials; that is, a digested method with 
general rules." 

This controversy is to be regretted on account of the long 
and bitter alienation which it produced between English and 
Continental mathematicians. It stopped almost completely 
all interchange of ideas on scientific subjects. The English 
adhered closely to Newton's methods and, until about 1820, 
remained, in most cases, ignorant of the brilliant mathematical 
discoveries that were being made on the Continent. The loss 
in point of scientific advantage was almost entirely on the 
side of Britain. The only way in which this dispute may 


be said, in a small measure, to have furthered the progress 
of mathematics, is through the challenge problems by which 
each side attempted to annoy its adversaries. 

The recurring practice of issuing challenge problems was 
inaugurated at this time by Leibniz. They were, at first, 
not intended as defiances, but merely as exercises in the new 
calculus. Such was the problem of the isochronous curve 
(to find the curve along which a body falls with uniform 
velocity), proposed by him to the Cartesians in 1687, and 
solved by James Bernoulli, himself, and John Bernoulli. James 
Bernoulli proposed in the Leipzig Journal the question to find 
the curve (the catenary) formed by a chain of uniform weight 
suspended freely from its ends. It was resolved by Huygens, 
Leibniz, and himself. In 1697 John Bernoulli challenged the 
best mathematicians in Europe to solve the difficult problem, 
to find the curve (the cycloid) along which a body falls from 
one point to another in the shortest possible time. Leibniz 
solved it the day he received it. Newton, de l'Hospital, 
and the two Bernoullis gave solutions. Newton's appeared 
anonymously in the Philosophical Transactions, but John 
Bernoulli recognised in it his powerful mind, "anquam," 
he says, "ex ungue leonem." The problem of orthogonal 
trajectories (a system of curves described by a known law 
being given, to describe a curve which shall cut them all at 
right angles) had been long proposed in the Acta Eruditorum, 
but failed at first to receive much attention. It was again 
proposed in 1716 by Leibniz, to feel the pulse of the English 


This may be considered as the first defiance problem 
professedly aimed at the English. Newton solved it the same 
evening on which it was delivered to him, although he was 
much fatigued by the day's work at the mint. His solution, as 
published, was a general plan of an investigation rather than 
an actual solution, and was, on that account, criticised by 
Bernoulli as being of no value. Brook Taylor undertook the 
defence of it, but ended by using very reprehensible language. 
Bernoulli was not to be outdone in incivility, and made a bitter 
reply. Not long afterwards Taylor sent an open defiance to 
Continental mathematicians of a problem on the integration 
of a fluxion of complicated form which was known to very 
few geometers in England and supposed to be beyond the 
power of their adversaries. The selection was injudicious, 
for Bernoulli had long before explained the method of this 
and similar integrations. It served only to display the skill 
and augment the triumph of the followers of Leibniz. The 
last and most unskilful challenge was by John Keill. The 
problem was to find the path of a projectile in a medium 
which resists proportionally to the square of the velocity. 
Without first making sure that he himself could solve it, 
Keill boldly challenged Bernoulli to produce a solution. The 
latter resolved the question in very short time, not only for 
a resistance proportional to the square, but to any power of 
the velocity. Suspecting the weakness of the adversary, he 
repeatedly offered to send his solution to a confidential person 
in London, provided Keill would do the same. Keill never 
made a reply, and Bernoulli abused him and cruelly exulted 


over him. [26] 

The explanations of the fundamental principles of the 
calculus, as given by Newton and Leibniz, lacked clearness 
and rigour. For that reason it met with opposition from several 
quarters. In 1694 Bernard Nieuwentyt of Holland denied the 
existence of differentials of higher orders and objected to 
the practice of neglecting infinitely small quantities. These 

objections Leibniz was not able to meet satisfactorily. In his 

reply he said the value of — in geometry could be expressed 

as the ratio of finite quantities. In the interpretation of 

dx and dy Leibniz vacillated. At one time they appear in his 

writings as finite lines; then they are called infinitely small 

quantities, and again, quantitates inassignabiles, which spring 

from quantitates assignabiles by the law of continuity. In this 

last presentation Leibniz approached nearest to Newton. 

In England the principles of fluxions were boldly attacked 

by Bishop Berkeley, the eminent metaphysician, who argued 

with great acuteness, contending, among other things, that 

the fundamental idea of supposing a finite ratio to exist 

between terms absolutely evanescent — "the ghosts of departed 

quantities," as he called them — was absurd and unintelligible. 

The reply made by Jurin failed to remove all the objections. 

Berkeley was the first to point out what was again shown 

later by Lazare Carnot, that correct answers were reached by 

a "compensation of errors." Berkeley's attack was not devoid 

of good results, for it was the immediate cause of the work on 

fluxions by Maclaurin. In France Michel Rolle rejected the 

differential calculus and had a controversy with Varignon on 


the subject. 

Among the most vigorous promoters of the calculus on the 
Continent were the Bernoullis. They and Euler made Basel 
in Switzerland famous as the cradle of great mathematicians. 
The family of Bernoullis furnished in course of a century eight 
members who distinguished themselves in mathematics. We 
subjoin the following genealogical table:— 

Nicolaus Bernoulli, the Father 

Jacob, 1654-1705 Nicolaus Johann, 1667-1748 

I I 

Nicolaus, 1687-1759 Nicolaus, 1695-1726 
Daniel, 1700-1782 
Johann, 1710-1790 

Daniel Johann, 1744-1807 Jacob, 1758-1789 

Most celebrated were the two brothers Jacob (James) and 
Johann (John), and Daniel, the son of John. James and John 
were staunch friends of Leibniz and worked hand in hand 
with him. James Bernoulli (1654-1705) was born in Basel. 
Becoming interested in the calculus, he mastered it without 
aid from a teacher. From 1687 until his death he occupied the 
mathematical chair at the University of Basel. He was the 
first to give a solution to Leibniz's problem of the isochronous 
curve. In his solution, published in the Acta Eruditorum, 
1690, we meet for the first time with the word integral. Leibniz 
had called the integral calculus calculus summatorius, but in 
1696 the term calculus integralis was agreed upon between 
Leibniz and John Bernoulli. James proposed the problem 


of the catenary, then proved the correctness of Leibniz's 
construction of this curve, and solved the more complicated 
problems, supposing the string to be (1) of variable density, 
(2) extensible, (3) acted upon at each point by a force directed 
to a fixed centre. Of these problems he published answers 
without explanations, while his brother John gave in addition 
their theory. He determined the shape of the "elastic curve" 
formed by an elastic plate or rod fixed at one end and bent 
by a weight applied to the other end; of the "lintearia," a 
flexible rectangular plate with two sides fixed horizontally 
at the same height, filled with a liquid; of the "volaria," a 
rectangular sail filled with wind. He studied the loxodromic 
and logarithmic spirals, in the last of which he took particular 
delight from its remarkable property of reproducing itself 
under a variety of conditions. Following the example of 
Archimedes, he willed that the curve be engraved upon his 
tomb-stone with the inscription u eadem mutata resurgo." 
In 1696 he proposed the famous problem of isoperimetrical 
figures, and in 1701 published his own solution. He wrote 
a work on Ars Conjectandi, which is a development of the 
calculus of probabilities and contains the investigation now 
called "Bernoulli's theorem" and the so-called "numbers of 
Bernoulli," which are in fact (though not so considered by 
him) the coefficients of — - in the expansion of (e x — 1) _1 . Of 
his collected works, in three volumes, one was printed in 1713, 
the other two in 1744. 

John Bernoulli (1667-1748) was initiated into mathe- 
matics by his brother. He afterwards visited France, where 


he met Malebranche, Cassini, De Lahire, Varignon, and de 
l'Hospital. For ten years he occupied the mathematical chair 
at Groningen and then succeeded his brother at Basel. He 
was one of the most enthusiastic teachers and most successful 
original investigators of his time. He was a member of almost 
every learned society in Europe. His controversies were almost 
as numerous as his discoveries. He was ardent in his friend- 
ships, but unfair, mean, and violent toward all who incurred 
his dislike — even his own brother and son. He had a bitter 
dispute with James on the isoperimetrical problem. James 
convicted him of several paralogisms. After his brother's 
death he attempted to substitute a disguised solution of the 
former for an incorrect one of his own. John admired the 
merits of Leibniz and Euler, but was blind to those of Newton. 
He immensely enriched the integral calculus by his labours. 
Among his discoveries are the exponential calculus, the line 
of swiftest descent, and its beautiful relation to the path 
described by a ray passing through strata of variable density. 
He treated trigonometry by the analytical method, studied 
caustic curves and trajectories. Several times he was given 
prizes by the Academy of Science in Paris. 

Of his sons, Nicholas and Daniel were appointed pro- 
fessors of mathematics at the same time in the Academy of 
St. Petersburg. The former soon died in the prime of life; the 
latter returned to Basel in 1733, where he assumed the chair of 
experimental philosophy. His first mathematical publication 
was the solution of a differential equation proposed by Riccati. 
He wrote a work on hydrodynamics. His investigations on 


probability are remarkable for their boldness and originality. 
He proposed the theory of moral expectation, which he thought 
would give results more in accordance with our ordinary no- 
tions than the theory of mathematical probability. His "moral 
expectation" has become classic, but no one ever makes use 
of it. He applies the theory of probability to insurance; to 
determine the mortality caused by small-pox at various stages 
of life; to determine the number of survivors at a given age from 
a given number of births; to determine how much inoculation 
lengthens the average duration of life. He showed how the 
differential calculus could be used in the theory of probability. 
He and Euler enjoyed the honour of having gained or shared 
no less than ten prizes from the Academy of Sciences in Paris. 

Johann Bernoulli (born 1710) succeeded his father in the 
professorship of mathematics at Basel. He captured three 
prizes (on the capstan, the propagation of light, and the 
magnet) from the Academy of Sciences at Paris. Nicolaus 
Bernoulli (born 1687) held for a time the mathematical chair 
at Padua which Galileo had once filled. Johann Bernoulli 
(born 1744) at the age of nineteen was appointed astronomer 
royal at Berlin, and afterwards director of the mathematical 
department of the Academy. His brother Jacob took upon 
himself the duties of the chair of experimental physics at 
Basel, previously performed by his uncle Jacob, and later 
was appointed mathematical professor in the Academy at St. 

Brief mention will now be made of some other mathemati- 
cians belonging to the period of Newton, Leibniz, and the 


elder Bernoullis. 

Guillaume Frangois Antoine l'Hospital (1661-1704), a 
pupil of John Bernoulli, has already been mentioned as taking 
part in the challenges issued by Leibniz and the Bernoullis. 
He helped powerfully in making the calculus of Leibniz better 
known to the mass of mathematicians by the publication of a 
treatise thereon in 1696. This contains for the first time the 
method of finding the limiting value of a fraction whose two 
terms tend toward zero at the same time. 

Another zealous French advocate of the calculus was Pierre 
Varignon (1654-1722). Joseph Saurin (1659-1737) solved 
the delicate problem of how to determine the tangents at 
the multiple points of algebraic curves. Frangois Nicole 
(1683-1758) in 1717 issued the first systematic treatise on 
finite differences, in which he finds the sums of a considerable 
number of interesting series. He wrote also on roulettes, 
particularly spherical epicycloids, and their rectification. 
Also interested in finite differences was Pierre Raymond 
de Montmort (1678-1719). His chief writings, on the theory 
of probability, served to stimulate his more distinguished 
successor, De Moivre. Jean Paul de Gua (1713-1785) 
gave the demonstration of Descartes' rule of signs, now 
given in books. This skilful geometer wrote in 1740 a work on 
analytical geometry, the object of which was to show that most 
investigations on curves could be carried on with the analysis 
of Descartes quite as easily as with the calculus. He shows 
how to find the tangents, asymptotes, and various singular 
points of curves of all degrees, and proved by perspective that 


several of these points can be at infinity. A mathematician 
who clung to the methods of the ancients was Philippe 
de Lahire (1640-1718), a pupil of Desargues. His work on 
conic sections is purely synthetic, but differs from ancient 
treatises in deducing the properties of conies from those of 
the circle in the same manner as did Desargues and Pascal. 
His innovations stand in close relation with modern synthetic 
geometry. He wrote on roulettes, on graphical methods, 
epicycloids, conchoids, and on magic squares. Michel Rolle 
(1652-1719) is the author of a theorem named after him. 

Of Italian mathematicians, Riccati and Fagnano must not 
remain unmentioned. Jacopo Francesco, Count Riccati 
(1676-1754) is best known in connection with his problem, 
called Riccati's equation, published in the Acta Eruditorum 
in 1724. He succeeded in integrating this differential equation 
for some special cases. A geometrician of remarkable power 

was Giulio Carlo, Count de Fagnano (1682-1766). He 

1 — i 

discovered the following formula, n = 2ilog , in which 

1 +i 
he anticipated Euler in the use of imaginary exponents and 

logarithms. His studies on the rectification of the ellipse 
and hyperbola are the starting-points of the theory of elliptic 
functions. He showed, for instance, that two arcs of an ellipse 
can be found in an indefinite number of ways, whose difference 
is expressible by a right line. 

In Germany the only noted contemporary of Leibniz is 
Ehrenfried Walter Tschirnhausen (1651-1708), who dis- 
covered the caustic of reflection, experimented on metallic 
reflectors and large burning-glasses, and gave us a method of 


transforming equations named after him. Believing that the 
most simple methods (like those of the ancients) are the most 
correct, he concluded that in the researches relating to the 
properties of curves the calculus might as well be dispensed 

After the death of Leibniz there was in Germany not a 
single mathematician of note. Christian Wolf (1679-1754), 
professor at Halle, was ambitious to figure as successor of 
Leibniz, but he "forced the ingenious ideas of Leibniz into a 
pedantic scholasticism, and had the unenviable reputation of 
having presented the elements of the arithmetic, algebra, and 
analysis developed since the time of the Renaissance in the 
form of Euclid, — of course only in outward form, for into the 
spirit of them he was quite unable to penetrate." [16] 

The contemporaries and immediate successors of Newton in 
Great Britain were men of no mean merit. We have reference 
to Cotes, Taylor, Maclaurin, and De Moivre. We are told that 
at the death of Roger Cotes (1682-1716), Newton exclaimed, 
"If Cotes had lived, we might have known something." It 
was at the request of Dr. Bentley that Cotes undertook the 
publication of the second edition of Newton's Principia. His 
mathematical papers were published after his death by Robert 
Smith, his successor in the Plumbian professorship at Trinity 
College. The title of the work, Harmonia Mensurarum, was 
suggested by the following theorem contained in it: If on 
each radius vector, through a fixed point O, there be taken 
a point R, such that the reciprocal of OR be the arithmetic 
mean of the reciprocals of OR±, OR2, ■ ■ ■ OR n , then the locus 


of R will be a straight line. In this work progress was made 
in the application of logarithms and the properties of the 
circle to the calculus of fluents. To Cotes we owe a theorem 
in trigonometry which depends on the forming of factors 
of x n — 1 . Chief among the admirers of Newton were Taylor 
and Maclaurin. The quarrel between English and Continental 
mathematicians caused them to work quite independently of 
their great contemporaries across the Channel. 

BrookTaylor (1685-1731) was interested inmany branches 
of learning, and in the latter part of his life engaged mainly 
in religious and philosophic speculations. His principal work, 
Methodus incrementorum directa et inversa, London, 1715— 
1717, added a new branch to mathematics, now called "finite 
differences." He made many important applications of it, 
particularly to the study of the form of movement of vibrating 
strings, first reduced to mechanical principles by him. This 
work contains also "Taylor's theorem," the importance of 
which was not recognised by analysts for over fifty years, 
until Lagrange pointed out its power. His proof of it does 
not consider the question of convergency, and is quite worth- 
less. The first rigorous proof was given a century later by 
Cauchy. Taylor's work contains the first correct explanation 
of astronomical refraction. He wrote also a work on linear 
perspective, a treatise which, like his other writings, suffers 
for want of fulness and clearness of expression. At the age 
of twenty-three he gave a remarkable solution of the problem 
of the centre of oscillation, published in 1714. His claim to 
priority was unjustly disputed by John Bernoulli. 


Colin Maclaurin (1698-1746) was elected professor of 

mathematics at Aberdeen at the age of nineteen by competitive 

examination, and in 1725 succeeded James Gregory at the 

University of Edinburgh. He enjoyed the friendship of Newton, 

and, inspired by Newton's discoveries, he published in 1719 

his Geometria Organica, containing a new and remarkable 

mode of generating conies, known by his name. A second 

tract, De Linearum geometricarum Proprietatibus, 1720, is 

remarkable for the elegance of its demonstrations. It is based 

upon two theorems: the first is the theorem of Cotes; the 

second is Maclaurin's: If through any point O a line be 

drawn meeting the curve in n points, and at these points 

tangents be drawn, and if any other line through O cut the 

curve in R\, i?2, etc., and the system of n tangents in n, 

ro, etc., then Y] — — = Y] ——. This and Cotes' theorem are 
Z} ' ^ OR ^ Or 

generalisations of theorems of Newton. Maclaurin uses these 
in his treatment of curves of the second and third degree, 
culminating in the remarkable theorem that if a quadrangle 
has its vertices and the two points of intersection of its 
opposite sides upon a curve of the third degree, then the 
tangents drawn at two opposite vertices cut each other on 
the curve. He deduced independently Pascal's theorem on 
the hexagram. The following is his extension of this theorem 
{Phil. Trans., 1735): If a polygon move so that each of its sides 
passes through a fixed point, and if all its summits except 
one describe curves of the degrees m, n, p, etc., respectively, 
then the free summit moves on a curve of the degree 2mnp • • • , 
which reduces to mnp ■ ■ ■ when the fixed points all lie on 


a straight line. Maclaurin wrote on pedal curves. He is 
the author of an Algebra. The object of his treatise on 
Fluxions was to found the doctrine of fluxions on geometric 
demonstrations after the manner of the ancients, and thus, by 
rigorous exposition, answer such attacks as Berkeley's that the 
doctrine rested on false reasoning. The Fluxions contained 
for the first time the correct way of distinguishing between 
maxima and minima, and explained their use in the theory of 
multiple points. "Maclaurin's theorem" was previously given 
by James Stirling, and is but a particular case of "Taylor's 
theorem." Appended to the treatise on Fluxions is the 
solution of a number of beautiful geometric, mechanical, and 
astronomical problems, in which he employs ancient methods 
with such consummate skill as to induce Clairaut to abandon 
analytic methods and to attack the problem of the figure 
of the earth by pure geometry. His solutions commanded 
the liveliest admiration of Lagrange. Maclaurin investigated 
the attraction of the ellipsoid of revolution, and showed that 
a homogeneous liquid mass revolving uniformly around an 
axis under the action of gravity must assume the form of 
an ellipsoid of revolution. Newton had given this theorem 
without proof. Notwithstanding the genius of Maclaurin, his 
influence on the progress of mathematics in Great Britain was 
unfortunate; for, by his example, he induced his countrymen 
to neglect analysis and to be indifferent to the wonderful 
progress in the higher analysis made on the Continent. 

It remains for us to speak of Abraham de Moivre (1667- 
1754), who was of French descent, but was compelled to leave 


France at the age of eighteen, on the Revocation of the Edict 
of Nantes. He settled in London, where he gave lessons in 
mathematics. He lived to the advanced age of eighty-seven 
and sank into a state of almost total lethargy. His subsistence 
was latterly dependent on the solution of questions on games 
of chance and problems on probabilities, which he was in the 
habit of giving at a tavern in St. Martin's Lane. Shortly 
before his death he declared that it was necessary for him 
to sleep ten or twenty minutes longer every day. The day 
after he had reached the total of over twenty-three hours, 
he slept exactly twenty-four hours and then passed away 
in his sleep. De Moivre enjoyed the friendship of Newton 
and H alley. His power as a mathematician lay in analytic 
rather than geometric investigation. He revolutionised higher 
trigonometry by the discovery of the theorem known by his 
name and by extending the theorems on the multiplication 
and division of sectors from the circle to the hyperbola. His 
work on the theory of probability surpasses anything done 
by any other mathematician except Laplace. His principal 
contributions are his investigations respecting the Duration 
of Play, his Theory of Recurring Series, and his extension 
of the value of Bernoulli's theorem by the aid of Stirling's 
theorem. [42] His chief works are the Doctrine of Chances, 
1716, the Miscellanea Analytica, 1730, and his papers in the 
Philosophical Transactions. 



During the epoch of ninety years from 1730 to 1820 the 
French and Swiss cultivated mathematics with most brilliant 
success. No previous period had shown such an array of 
illustrious names. At this time Switzerland had her Euler; 
France, her Lagrange, Laplace, Legendre, and Monge. The 
mediocrity of French mathematics which marked the time of 
Louis XIV. was now followed by one of the very brightest 
periods of all history. England and Germany, on the other 
hand, which during the unproductive period in France had 
their Newton and Leibniz, could now boast of no great 
mathematician. France now waved the mathematical sceptre. 
Mathematical studies among the English and German people 
had sunk to the lowest ebb. Among them the direction 
of original research was ill-chosen. The former adhered 
with excessive partiality to ancient geometrical methods; the 
latter produced the combinatorial school, which brought forth 
nothing of value. 

The labours of Euler, Lagrange, and Laplace lay in higher 
analysis, and this they developed to a wonderful degree. By 
them analysis came to be completely severed from geometry. 
During the preceding period the effort of mathematicians 
not only in England, but, to some extent, even on the 
continent, had been directed toward the solution of problems 
clothed in geometric garb, and the results of calculation 
were usually reduced to geometric form. A change now 
took place. Euler brought about an emancipation of the 


analytical calculus from geometry and established it as an 
independent science. Lagrange and Laplace scrupulously 
adhered to this separation. Building on the broad foundation 
laid for higher analysis and mechanics by Newton and Leibniz, 
Euler, with matchless fertility of mind, erected an elaborate 
structure. There are few great ideas pursued by succeeding 
analysts which were not suggested by Euler, or of which 
he did not share the honour of invention. With, perhaps, 
less exuberance of invention, but with more comprehensive 
genius and profounder reasoning, Lagrange developed the 
infinitesimal calculus and put analytical mechanics into the 
form in which we now know it. Laplace applied the calculus 
and mechanics to the elaboration of the theory of universal 
gravitation, and thus, largely extending and supplementing 
the labours of Newton, gave a full analytical discussion of 
the solar system. He also wrote an epoch-marking work 
on Probability. Among the analytical branches created 
during this period are the calculus of Variations by Euler and 
Lagrange, Spherical Harmonics by Laplace and Legendre, and 
Elliptic Integrals by Legendre. 

Comparing the growth of analysis at this time with the 
growth during the time of Gauss, Cauchy, and recent math- 
ematicians, we observe an important difference. During the 
former period we witness mainly a development with refer- 
ence to form. Placing almost implicit confidence in results of 
calculation, mathematicians did not always pause to discover 
rigorous proofs, and were thus led to general propositions, 
some of which have since been found to be true in only special 


cases. The Combinatorial School in Germany carried this 
tendency to the greatest extreme; they worshipped formalism 
and paid no attention to the actual contents of formulae. 
But in recent times there has been added to the dexterity in 
the formal treatment of problems, a much-needed rigour of 
demonstration. A good example of this increased rigour is 
seen in the present use of infinite series as compared to that of 
Euler, and of Lagrange in his earlier works. 

The ostracism of geometry, brought about by the master- 
minds of this period, could not last permanently. Indeed, a 
new geometric school sprang into existence in France before 
the close of this period. Lagrange would not permit a single 
diagram to appear in his Mecanique analytique, but thirteen 
years before his death, Monge published his epoch-making 
Geometrie descriptive. 

Leonhard Euler (1707-1783) was born in Basel. His 
father, a minister, gave him his first instruction in mathematics 
and then sent him to the University of Basel, where he became 
a favourite pupil of John Bernoulli. In his nineteenth year 
he composed a dissertation on the masting of ships, which 
received the second prize from the French Academy of Sciences. 
When John Bernoulli's two sons, Daniel and Nicolaus, went 
to Russia, they induced Catharine I., in 1727, to invite their 
friend Euler to St. Petersburg, where Daniel, in 1733, was 
assigned to the chair of mathematics. In 1735 the solving of an 
astronomical problem, proposed by the Academy, for which 
several eminent mathematicians had demanded some months' 
time, was achieved in three days by Euler with aid of improved 


methods of his own. But the effort threw him into a fever and 
deprived him of the use of his right eye. With still superior 
methods this same problem was solved later by the illustrious 
Gauss in one hour! [47] The despotism of Anne I. caused the 
gentle Euler to shrink from public affairs and to devote all 
his time to science. After his call to Berlin by Frederick the 
Great in 1747, the queen of Prussia, who received him kindly, 
wondered how so distinguished a scholar should be so timid 
and reticent. Euler naively replied, "Madam, it is because I 
come from a country where, when one speaks, one is hanged." 
In 1766 he with difficulty obtained permission to depart from 
Berlin to accept a call by Catharine II. to St. Petersburg. 
Soon after his return to Russia he became blind, but this 
did not stop his wonderful literary productiveness, which 
continued for seventeen years, until the day of his death. [45] 
He dictated to his servant his Anleitung zur Algebra, 1770, 
which, though purely elementary, is meritorious as one of the 
earliest attempts to put the fundamental processes on a sound 

Euler wrote an immense number of works, chief of which are 
thefollowing: Introductio in analysininfinitorum, 1748,awork 
that caused a revolution in analytical mathematics, a subject 
which had hitherto never been presented in so general and 
systematic manner; Institutiones calculi differential, 1755, 
and Institutiones calculi integralis, 1768-1770, which were the 
most complete and accurate works on the calculus of that time, 
and contained not only a full summary of everything then 
known on this subject , but also the Beta and Gamma Functions 


and other original investigations; Methodus inveniendi lineas 
curvas maximi minimive proprietate gaudentes, 1744, which, 
displaying an amount of mathematical genius seldom rivalled, 
contained his researches on the calculus of variations (a 
subject afterwards improved by Lagrange), to the invention 
of which Euler was led by the study of isoperimetrical curves, 
the brachistochrone in a resisting medium, and the theory 
of geodesies (subjects which had previously engaged the 
attention of the elder Bernoullis and others); the Theoria 
motuum planetarum et cometarum, 1744, Theoria motus 
lun<E, 1753, Theoria motuum Iutkb, 1772, are his chief works 
on astronomy; Ses lettres a une princesse d'Allemagne sur 
quelques sujets de Physique et de Philosophie, 1770, was a 
work which enjoyed great popularity. 

We proceed to mention the principal innovations and 
inventions of Euler. He treated trigonometry as a branch of 
analysis, introduced (simultaneously with Thomas Simpson 
in England) the now current abbreviations for trigonometric 
functions, and simplified formulae by the simple expedient of 
designating the angles of a triangle by A, B, C, and the opposite 
sides by a, b, c, respectively. He pointed out the relation 
between trigonometric and exponential functions. In a paper 
of 1737 we first meet the symbol n to denote 3.14159 . . .. [21] 
Euler laid down the rules for the transformation of co-ordinates 
in space, gave a methodic analytic treatment of plane curves 
and of surfaces of the second order. He was the first to discuss 
the equation of the second degree in three variables, and to 
classify the surfaces represented by it. By criteria analogous 


to those used in the classification of conies he obtained five 
species. He devised a method of solving biquadratic equations 
by assuming x = V / P+ \fq+ \Jr, with the hope that it would lead 
him to a general solution of algebraic equations. The method 
of elimination by solving a series of linear equations (invented 
independently by Bezout) and the method of elimination by 
symmetric functions, are due to him. [20] Far reaching are 
Euler's researches on logarithms. Leibniz and John Bernoulli 
once argued the question whether a negative number has 
a logarithm. Bernoulli claimed that since (—a) 2 = (+a) 2 , 
we have log(-a) 2 = log(+a) 2 and 21og(— a) = 21og(+a), and 
finally log(— a) = log(+a). Euler proved that a has really an 
infinite number of logarithms, all of which are imaginary when 
a is negative, and all except one when a is positive. He then 
explained how log(-a) 2 might equal log(+a) 2 , and yet log(-a) 
not equal log(+o). 

The subject of infinite series received new life from him. To 
his researches on series we owe the creation of the theory of 
definite integrals by the development of the so-called Eulerian 
integrals. He warns his readers occasionally against the use 
of divergent series, but is nevertheless very careless himself. 
The rigid treatment to which infinite series are subjected 
now was then undreamed of. No clear notions existed as to 
what constitutes a convergent series. Neither Leibniz nor 
Jacob and John Bernoulli had entertained any serious doubt 
of the correctness of the expression \ = 1 — 1 + 1 — 1 + ---. 
Guido Grandi went so far as to conclude from this that 
5 = + + 0H .In the treatment of series Leibniz advanced 


a metaphysical method of proof which held sway over the 
minds of the elder Bernoullis, and even of Euler. [46] The 
tendency of that reasoning was to justify results which seem 
to us now highly absurd. The looseness of treatment can 
best be seen from examples. The very paper in which Euler 
cautions against divergent series contains the proof that 

k H hl + n + n 2 H = 0as follows: 

n z n 

9 n 11 n 

n + n z + ■•■=- , l+- + ^r + 

1 — n ' n n 2 n — 1 ' 

these added give zero. Euler has no hesitation to write 
1 — 3 + 5 — 7 + • • • = 0, and no one objected to such results 
excepting Nicolaus Bernoulli, the nephew of John and Jacob. 
Strange to say, Euler finally succeeded in converting Nicolaus 
Bernoulli to his own erroneous views. At the present time it is 
difficult to believe that Euler should have confidently written 
sin <j> — 2 sin 2(p + 3 sin 30 — 4 sin 40 + • • • = 0, but such examples 
afford striking illustrations of the want of scientific basis of 
certain parts of analysis at that time. Euler's proof of the 
binomial formula for negative and fractional exponents, which 
has been reproduced in elementary text-books of even recent 
years, is faulty. A remarkable development, due to Euler, 
is what he named the hypergeometric series, the summation 
of which he observed to be dependent upon the integration 
of a linear differential equation of the second order, but it 
remained for Gauss to point out that for special values of 
its letters, this series represented nearly all functions then 

Euler developed the calculus of finite differences in the first 


chapters of his Institutiones calculi differentialis, and then 
deduced the differential calculus from it. He established a 
theorem on homogeneous functions, known by his name, and 
contributed largely to the theory of differential equations, a 
subject which had received the attention of Newton, Leibniz, 
and the Bernoullis, but was still undeveloped. Clairaut, 
Fontaine, and Euler about the same time observed criteria 
of integrability, but Euler in addition showed how to employ 
them to determine integrating factors. The principles on 
which the criteria rested involved some degree of obscurity. 
The celebrated addition-theorem for elliptic integrals was 
first established by Euler. He invented a new algorithm for 
continued fractions, which he employed in the solution of 
the indeterminate equation ax + by = c. We now know that 
substantially the same solution of this equation was given 
1000 years earlier, by the Hindoos. By giving the factors 
of the number 2 2 + 1 when n = 5, he pointed out that this 
expression did not always represent primes, as was supposed 
by Fermat. He first supplied the proof to "Fermat's theorem," 
and to a second theorem of Fermat, which states that every 
prime of the form 4n + 1 is expressible as the sum of two 
squares in one and only one way. A third theorem of Fermat, 
that x n + y n = z n , has no integral solution for values of n 
greater than 2, was proved by Euler to be correct when n = 3. 
Euler discovered four theorems which taken together make 
out the great law of quadratic reciprocity, a law independently 
discovered by Legendre. [48] Euler enunciated and proved a 
well-known theorem, giving the relation between the number 


of vertices, faces, and edges of certain polyhedra, which, 
however, appears to have been known to Descartes. The 
powers of Euler were directed also towards the fascinating 
subject of the theory of probability, in which he solved some 
difficult problems. 

Of no little importance are Euler's labours in analytical 
mechanics. Says Whewell: "The person who did most to give 
to analysis the generality and symmetry which are now its 
pride, was also the person who made mechanics analytical; I 
mean Euler." [11] He worked out the theory of the rotation 
of a body around a fixed point, established the general 
equations of motion of a free body, and the general equation of 
hydrodynamics. He solved an immense number and variety of 
mechanical problems, which arose in his mind on all occasions. 
Thus, on reading Virgil's lines, "The anchor drops, the rushing 
keel is staid," he could not help inquiring what would be the 
ship's motion in such a case. About the same time as Daniel 
Bernoulli he published the Principle of the Conservation of 
Areas and defended the principle of "least action," advanced 
by Maupertius. He wrote also on tides and on sound. 

Astronomy owes to Euler the method of the variation 
of arbitrary constants. By it he attacked the problem of 
perturbations, explaining, in case of two planets, the secular 
variations of eccentricities, nodes, etc. He was one of the 
first to take up with success the theory of the moon's motion 
by giving approximate solutions to the "problem of three 
bodies." He laid a sound basis for the calculation of tables 
of the moon. These researches on the moon's motion, which 


captured two prizes, were carried on while he was blind, with 
the assistance of his sons and two of his pupils. 

Most of his memoirs are contained in the transactions of 
the Academy of Sciences at St. Petersburg, and in those of 
the Academy at Berlin. From 1728 to 1783 a large portion 
of the Petropolitan transactions were filled by his writings. 
He had engaged to furnish the Petersburg Academy with 
memoirs in sufficient number to enrich its acts for twenty 
years — a promise more than fulfilled, for down to 1818 the 
volumes usually contained one or more papers of his. It has 
been said that an edition of Euler's complete works would 
fill 16, 000 quarto pages. His mode of working was, first 
to concentrate his powers upon a special problem, then to 
solve separately all problems growing out of the first. No 
one excelled him in dexterity of accommodating methods to 
special problems. It is easy to see that mathematicians could 
not long continue in Euler's habit of writing and publishing. 
The material would soon grow to such enormous proportions 
as to be unmanageable. We are not surprised to see almost 
the opposite in Lagrange, his great successor. The great 
Frenchman delighted in the general and abstract, rather than, 
like Euler, in the special and concrete. His writings are 
condensed and give in a nutshell what Euler narrates at great 

Jean-le-Rond D'Alembert (1717 1783) was exposed, 
when an infant, by his mother in a market by the church of St. 
Jean-le-Rond, near the Notre-Dame in Paris, from which he 
derived his Christian name. He was brought up by the wife of a 


poor glazier. It is said that when he began to show signs of great 
talent, his mother sent for him, but received the reply, "You 
are only my step-mother; the glazier's wife is my mother." His 
father provided him with a yearly income. D' Alembert entered 
upon the study of law, but such was his love for mathematics, 
that law was soon abandoned. At the age of twenty- four 
his reputation as a mathematician secured for him admission 
to the Academy of Sciences. In 1743 appeared his Traite 
de dynamique, founded upon the important general principle 
bearing his name: The impressed forces are equivalent to 
the effective forces. D'Alembert's principle seems to have 
been recognised before him by Fontaine, and in some measure 
by John Bernoulli and Newton. D'Alembert gave it a clear 
mathematical form and made numerous applications of it. It 
enabled the laws of motion and the reasonings depending on 
them to be represented in the most general form, in analytical 
language. D'Alembert applied it in 1744 in a treatise on 
the equilibrium and motion of fluids, in 1746 to a treatise 
on the general causes of winds, which obtained a prize from 
the Berlin Academy. In both these treatises, as also in one 
of 1747, discussing the famous problem of vibrating chords, 
he was led to partial differential equations. He was a leader 

among the pioneers in the study of such equations. To the 

d 2 y d 2 y 

equation — ^ = a 2 — ^, arising in the problem of vibrating 

chords, he gave as the general solution, 

y = f(x + at) + 4>(x - at), 
and showed that there is only one arbitrary function, if y be 


supposed to vanish for x = and x = I. Daniel Bernoulli, 
starting with a particular integral given by Brook Taylor, 
showed that this differential equation is satisfied by the 
trigonometric series 

nx nt lixx 2nt 

y = a sin — • cos — + p sin — — ■ cos — - — V ■ ■ ■ , 

and claimed this expression to be the most general solution. 
Euler denied its generality, on the ground that, if true, 
the doubtful conclusion would follow that the above series 
represents any arbitrary function of a variable. These doubts 
were dispelled by Fourier. Lagrange proceeded to find the sum 
of the above series, but D'Alembert rightly objected to his 
process, on the ground that it involved divergent series. [46] 

A most beautiful result reached by D'Alembert, with aid of 
his principle, was the complete solution of the problem of the 
precession of the equinoxes, which had baffled the talents of 
the best minds. He sent to the French Academy in 1747, on 
the same day with Clairaut, a solution of the problem of three 
bodies. This had become a question of universal interest to 
mathematicians, in which each vied to outdo all others. The 
problem of two bodies, requiring the determination of their 
motion when they attract each other with forces inversely 
proportional to the square of the distance between them, 
had been completely solved by Newton. The "problem of 
three bodies" asks for the motion of three bodies attracting 
each other according to the law of gravitation. Thus far, 
the complete solution of this has transcended the power of 
analysis. The general differential equations of motion were 


stated by Laplace, but the difficulty arises in their integration. 
The "solutions" hitherto given are merely convenient methods 
of approximation in special cases when one body is the sun, 
disturbing the motion of the moon around the earth, or where 
a planet moves under the influence of the sun and another 

In the discussion of the meaning of negative quantities, 
of the fundamental processes of the calculus, and of the 
theory of probability, D'Alembert paid some attention to the 
philosophy of mathematics. His criticisms were not always 
happy. In 1754 he was made permanent secretary of the 
French Academy. During the last years of his life he was 
mainly occupied with the great French encyclopaedia, which 
was begun by Diderot and himself. D'Alembert declined, in 
1762, an invitation of Catharine II. to undertake the education 
of her son. Frederick the Great pressed him to go to Berlin. 
He made a visit, but declined a permanent residence there. 

Alexis Claude Clairaut (1713-1765) was a youthful 
prodigy. He read l'Hospital's works on the infinitesimal 
calculus and on conic sections at the age of ten. In 1731 was 
published his Recherches sur les courbes a double courbure, 
which he had ready for the press when he was sixteen. It was 
a work of remarkable elegance and secured his admission to 
the Academy of Sciences when still under legal age. In 1731 
he gave a proof of the theorem enunciated by Newton, that 
every cubic is a projection of one of five divergent parabolas. 
Clairaut formed the acquaintance of Maupertius, whom he 
accompanied on an expedition to Lapland to measure the 


length of a degree of the meridian. At that time the shape 
of the earth was a subject of serious disagreement. Newton 
and Huygens had concluded from theory that the earth was 
flattened at the poles. About 1713 Dominico Cassini measured 
an arc extending from Dunkirk to Perpignan and arrived at 
the startling result that the earth is elongated at the poles. 
To decide between the conflicting opinions, measurements 
were renewed. Maupertius earned by his work in Lapland the 
title of "earth flattener" by disproving the Cassinian tenet 
that the earth was elongated at the poles, and showing that 
Newton was right. On his return, in 1743, Clairaut published 
a work, Theorie de la figure de la Terre, which was based 
on the results of Maclaurin on homogeneous ellipsoids. It 
contains a remarkable theorem, named after Clairaut, that 
the sum of the fractions expressing the ellipticity and the 
increase of gravity at the pole is equal to 2^ times the fraction 
expressing the centrifugal force at the equator, the unit of 
force being represented by the force of gravity at the equator. 
This theorem is independent of any hypothesis with respect 
to the law of densities of the successive strata of the earth. It 
embodies most of Clairaut's researches. Todhunter says that 
"in the figure of the earth no other person has accomplished 
so much as Clairaut, and the subject remains at present 
substantially as he left it, though the form is different. The 
splendid analysis which Laplace supplied, adorned but did 
not really alter the theory which started from the creative 
hands of Clairaut." 

In 1752 he gained a prize of the St. Petersburg Academy for 


his paper on Theorie de la Lune, in which for the first time 
modern analysis is applied to lunar motion. This contained 
the explanation of the motion of the lunar apsides. This 
motion, left unexplained by Newton, seemed to him at first 
inexplicable by Newton's law, and he was on the point of 
advancing a new hypothesis regarding gravitation, when, 
taking the precaution to carry his calculation to a higher 
degree of approximation, he reached results agreeing with 
observation. The motion of the moon was studied about the 
same time by Euler and D'Alembert. Clairaut predicted that 
"Halley's Comet," then expected to return, would arrive at 
its nearest point to the sun on April 13, 1759, a date which 
turned out to be one month too late. He was the first to detect 
singular solutions in differential equations of the first order 
but of higher degree than the first. 

In their scientific labours there was between Clairaut and 
D'Alembert great rivalry, often far from friendly. The growing 
ambition of Clairaut to shine in society, where he was a great 
favourite, hindered his scientific work in the latter part of his 

Johann Heinrich Lambert (1728-1777), born at Miihl- 
hausen in Alsace, was the son of a poor tailor. While 
working at his father's trade, he acquired through his own 
unaided efforts a knowledge of elementary mathematics. 
At the age of thirty he became tutor in a Swiss family 
and secured leisure to continue his studies. In his travels 
with his pupils through Europe he became acquainted with 
the leading mathematicians. In 1764 he settled in Berlin, 


where he became member of the Academy, and enjoyed 
the society of Euler and Lagrange. He received a small 
pension, and later became editor of the Berlin Ephemeris. 
His many-sided scholarship reminds one of Leibniz. In his 
Cosmological Letters he made some remarkable prophecies 
regarding the stellar system. In mathematics he made several 
discoveries which were extended and overshadowed by his 
great contemporaries. His first research on pure mathematics 
developed in an infinite series the root x of the equation 
x m _|_ p X _ g_ Since each equation of the form ax r + bx s = d 
can be reduced to x m + px = q in two ways, one or the other 
of the two resulting series was always found to be convergent, 
and to give a value of x. Lambert's results stimulated Euler, 
who extended the method to an equation of four terms, and 
particularly Lagrange, who found that a function of a root 
of a — x + 4>{x) = can be expressed by the series bearing 
his name. In 1761 Lambert communicated to the Berlin 
Academy a memoir, in which he proves that it is irrational. 
This proof is given in Note IV. of Legendre's Geometrie, where 
it is extended to -k 1 . To the genius of Lambert we owe the 
introduction into trigonometry of hyperbolic functions, which 
he designated by sinhx, coshx, etc. His Freye Perspective, 
1759 and 1773, contains researches on descriptive geometry, 
and entitle him to the honour of being the forerunner of 
Monge. In his effort to simplify the calculation of cometary 
orbits, he was led geometrically to some remarkable theorems 
on conies, for instance this: "If in two ellipses having a 
common major axis we take two such arcs that their chords 


are equal, and that also the sums of the radii vectores, drawn 
respectively from the foci to the extremities of these arcs, are 
equal to each other, then the sectors formed in each ellipse 
by the arc and the two radii vectores are to each other as the 
square roots of the parameters of the ellipses." [13] 

John Landen (1719-1790) was an English mathematician 
whose writings served as the starting-point of investigations by 
Euler, Lagrange, and Legendre. Landen's capital discovery, 
contained in a memoir of 1755, was that every arc of the 
hyperbola is immediately rectified by means of two arcs of 
an ellipse. In his "residual analysis" he attempted to obviate 
the metaphysical difficulties of fluxions by adopting a purely 
algebraic method. Lagrange's Calcul des Fonctions is based 
upon this idea. Landen showed how the algebraic expression 
for the roots of a cubic equation could be derived by application 
of the differential and integral calculus. Most of the time of 
this suggestive writer was spent in the pursuits of active life. 

Etienne Bezout (1730-1783) was a French writer of 
popular mathematical school-books. In his Theorie generate 
des Equations Algebriques, 1779, he gave the method of 
elimination by linear equations (invented also by Euler). This 
method was first published by him in a memoir of 1764, in 
which he uses determinants, without, however, entering upon 
their theory. A beautiful theorem as to the degree of the 
resultant goes by his name. 

Louis Arbogaste (1759-1803) of Alsace was professor of 
mathematics at Strasburg. His chief work, the Calcul des 
Derivations, 1800, gives the method known by his name, 


by which the successive coefficients of a development are 
derived from one another when the expression is complicated. 
De Morgan has pointed out that the true nature of derivation 
is differentiation accompanied by integration. In this book 
for the first time are the symbols of operation separated from 
those of quantity. The notation D x y for dy/dx is due to him. 

Maria Gaetana Agnesi (1718-1799) of Milan, distin- 
guished as a linguist, mathematician, and philosopher, filled 
the mathematical chair at the University of Bologna during 
her father's sickness. In 1748 she published her Instituzioni 
Analitiche, which was translated into English in 1801. The 

"witch of Agnesi" or "versiera" is a plane curve containing a 

V\ 2 , -, c 

straight line, x — 0, and a cubic I - ) +1 

Vc/ x 

Joseph Louis Lagrange (1736-1813), one of the greatest 
mathematicians of all times, was born at Turin and died at 
Paris. He was of French extraction. His father, who had 
charge of the Sardinian military chest, was once wealthy, 
but lost all he had in speculation. Lagrange considered this 
loss his good fortune, for otherwise he might not have made 
mathematics the pursuit of his life. While at the college in 
Turin his genius did not at once take its true bent. Cicero 
and Virgil at first attracted him more than Archimedes and 
Newton. He soon came to admire the geometry of the 
ancients, but the perusal of a tract of Halley roused his 
enthusiasm for the analytical method, in the development of 
which he was destined to reap undying glory. He now applied 
himself to mathematics, and in his seventeenth year he became 
professor of mathematics in the royal military academy at 


Turin. Without assistance or guidance he entered upon a 
course of study which in two years placed him on a level with 
the greatest of his contemporaries. With aid of his pupils he 
established a society which subsequently developed into the 
Turin Academy. In the first five volumes of its transactions 
appear most of his earlier papers. At the age of nineteen 
he communicated to Euler a general method of dealing with 
"isoperimetrical problems," known now as the Calculus of 
Variations. This commanded Euler's lively admiration, and 
he courteously withheld for a time from publication some 
researches of his own on this subject, so that the youthful 
Lagrange might complete his investigations and claim the 
invention. Lagrange did quite as much as Euler towards the 
creation of the Calculus of Variations. As it came from Euler 
it lacked an analytic foundation, and this Lagrange supplied. 
He separated the principles of this calculus from geometric 
considerations by which his predecessor had derived them. 
Euler had assumed as fixed the limits of the integral, i.e. the 
extremities of the curve to be determined, but Lagrange 
removed this restriction and allowed all co-ordinates of the 
curve to vary at the same time. Euler introduced in 1766 the 
name "calculus of variations," and did much to improve this 
science along the lines marked out by Lagrange. 

Another subject engaging the attention of Lagrange at Turin 
was the propagation of sound. In his papers on this subject 
in the Miscellanea Taurinensia, the young mathematician 
appears as the critic of Newton, and the arbiter between Euler 
and D'Alembert. By considering only the particles which are 


in a straight line, he reduced the problem to the same partial 
differential equation that represents the motions of vibrating 
strings. The general integral of this was found by D'Alembert 
to contain two arbitrary functions, and the question now 
came to be discussed whether an arbitrary function may be 
discontinuous. D'Alembert maintained the negative against 
Euler, Daniel Bernoulli, and finally Lagrange, — arguing that 
in order to determine the position of a point of the chord at a 
time t, the initial position of the chord must be continuous. 
Lagrange settled the question in the affirmative. 

By constant application during nine years, Lagrange, at 
the age of twenty-six, stood at the summit of European fame. 
But his intense studies had seriously weakened a constitution 
never robust, and though his physicians induced him to take 
rest and exercise, his nervous system never fully recovered its 
tone, and he was thenceforth subject to fits of melancholy. 

In 1764 the French Academy proposed as the subject of a 
prize the theory of the libration of the moon. It demanded 
an explanation, on the principle of universal gravitation, why 
the moon always turns, with but slight variations, the same 
face to the earth. Lagrange secured the prize. This success 
encouraged the Academy to propose as a prize the theory of 
the four satellites of Jupiter, — a problem of six bodies, more 
difficult than the one of three bodies previously solved by 
Clairaut, D'Alembert, and Euler. Lagrange overcame the 
difficulties, but the shortness of time did not permit him to 
exhaust the subject. Twenty-four years afterwards it was 
completed by Laplace. Later astronomical investigations of 


Lagrange are on cometary perturbations (1778 and 1783), 
on Kepler's problem, and on a new method of solving the 
problem of three bodies. 

Being anxious to make the personal acquaintance of leading 
mathematicians, Lagrange visited Paris, where he enjoyed the 
stimulating delight of conversing with Clairaut, D'Alembert, 
Condorcet, the Abbe Marie, and others. He had planned a 
visit to London, but he fell dangerously ill after a dinner in 
Paris, and was compelled to return to Turin. In 1766 Euler 
left Berlin for St. Petersburg, and he pointed out Lagrange 
as the only man capable of filling the place. D'Alembert 
recommended him at the same time. Frederick the Great 
thereupon sent a message to Turin, expressing the wish of "the 
greatest king of Europe" to have "the greatest mathematician" 
at his court. Lagrange went to Berlin, and staid there twenty 
years. Finding all his colleagues married, and being assured by 
their wives that the marital state alone is happy, he married. 
The union was not a happy one. His wife soon died. Frederick 
the Great held him in high esteem, and frequently conversed 
with him on the advantages of perfect regularity of life. This 
led Lagrange to cultivate regular habits. He worked no 
longer each day than experience taught him he could without 
breaking down. His papers were carefully thought out before 
he began writing, and when he wrote he did so without a 
single correction. 

During the twenty years in Berlin he crowded the trans- 
actions of the Berlin Academy with memoirs, and wrote 
also the epoch-making work called the Mecanique Analytique. 


He enriched algebra by researches on the solution of equa- 
tions. There are two methods of solving directly algebraic 
equations, — that of substitution and that of combination. The 
former method was developed by Ferrari, Vieta, Tchirnhausen, 
Euler, Bezout, and Lagrange; the latter by Vandermonde and 
Lagrange. [20] In the method of substitution the original 
forms are so transformed that the determination of the roots 
is made to depend upon simpler functions (resolvents). In the 
method of combination auxiliary quantities are substituted 
for certain simple combinations ("types") of the unknown 
roots of the equation, and auxiliary equations (resolvents) 
are obtained for these quantities with aid of the coefficients 
of the given equation. Lagrange traced all known algebraic 
solutions of equations to the uniform principle consisting in 
the formation and solution of equations of lower degree whose 
roots are linear functions of the required roots, and of the 
roots of unity. He showed that the quintic cannot be reduced 
in this way, its resolvent being of the sixth degree. His 
researches on the theory of equations were continued after 
he left Berlin. In the Resolution des equations numeriques 
(1798) he gave a method of approximating to the real roots 
of numerical equations by continued fractions. Among other 
things, it contains also a proof that every equation must 
have a root, — a theorem which appears before this to have 
been considered self-evident. Other proofs of this were given 
by Argand, Gauss, and Cauchy. In a note to the above 
work Lagrange uses Fermat's theorem and certain suggestions 
of Gauss in effecting a complete algebraic solution of any 


binomial equation. 

While in Berlin Lagrange published several papers on the 
theory of numbers. In 1769 he gave a solution in integers of 
indeterminate equations of the second degree, which resembles 
the Hindoo cyclic method; he was the first to prove, in 1771, 
"Wilson's theorem," enunciated by an Englishman, John 
Wilson, and first published by Waring in his Meditationes 
Algebraicte; he investigated in 1775 under what conditions 
±2 and ±5 (—1 and ±3 having been discussed by Euler) are 
quadratic residues, or non-residues of odd prime numbers, q; 
he proved in 1770 Meziriac's theorem that every integer is 
equal to the sum of four, or a less number, of squares. He 
proved Fermat's theorem on x n + y n = z n , for the case n = 4, 
also Fermat's theorem that, if a 2 + b 2 = c 2 , then ab is not a 

In his memoir on Pyramids, 1773, Lagrange made consider- 
able use of determinants of the third order, and demonstrated 
that the square of a determinant is itself a determinant. He 
never, however, dealt explicitly and directly with determi- 
nants; he simply obtained accidentally identities which are 
now recognised as relations between determinants. 

Lagrange wrote much on differential equations. Though 
the subject of contemplation by the greatest mathemati- 
cians (Euler, D'Alembert, Clairaut, Lagrange, Laplace), yet 
more than other branches of mathematics did they resist 
the systematic application of fixed methods and principles. 
Lagrange established criteria for singular solutions (Calcul 
des Fonctions, Lessons 14-17), which are, however, erroneous. 


He was the first to point out the geometrical significance of 
such solutions. He generalised Euler's researches on total 
differential equations of two variables, and of the ninth order; 
he gave a solution of partial differential equations of the first 
order (Berlin Memoirs, 1772 and 1774), and spoke of their 
singular solutions, extending their solution in Memoirs of 
1779 and 1785 to equations of any number of variables. The 
discussion on partial differential equations of the second order, 
carried on by D'Alembert, Euler, and Lagrange, has already 
been referred to in our account of D'Alembert. 

While in Berlin, Lagrange wrote the " Mecanique Analy- 
tique" the greatest of his works (Paris, 1788). From the 
principle of virtual velocities he deduced, with aid of the cal- 
culus of variations, the whole system of mechanics so elegantly 
and harmoniously that it may fitly be called, in Sir William 
Rowan Hamilton's words, "a kind of scientific poem." It is 
a most consummate example of analytic generality. Geo- 
metrical figures are nowhere allowed. "On ne trouvera point 
de figures dans cet ouvrage" (Preface). The two divisions 
of mechanics — statics and dynamics — are in the first four 
sections of each carried out analogously, and each is prefaced 
by a historic sketch of principles. Lagrange formulated the 
principle of least action. In their original form, the equations 
of motion involve the co-ordinates x, y, z, of the different 
particles m or dm of the system. But x, y, z, are in general 
not independent, and Lagrange introduced in place of them 
any variables £, ip, <j>, whatever, determining the position of 
the point at the time. These may be taken to be independent. 


The equations of motion may now assume the form 

d dT _dT „ _ 
dtd?~^ + ^~ ' 

or when H, *,$,.. . are the partial differential coefficients with 

respect to £, i/j, (f>, . . , of one and the same function V, then the 


d dT dT dV _ 

did^~^ + ^ ~ ' 
The latter is par excellence the Lagrangian form of the 
equations of motion. With Lagrange originated the remark 
that mechanics may be regarded as a geometry of four 
dimensions. To him falls the honour of the introduction of 
the potential into dynamics. [49] Lagrange was anxious to 
have his Mecanique Analytique published in Paris. The work 
was ready for print in 1786, but not till 1788 could he find a 
publisher, and then only with the condition that after a few 
years he would purchase all the unsold copies. The work was 
edited by Legendre. 

After the death of Frederick the Great, men of science were 
no longer respected in Germany, and Lagrange accepted an 
invitation of Louis XVI. to migrate to Paris. The French queen 
treated him with regard, and lodging was procured for him in 
the Louvre. But he was seized with a long attack of melancholy 
which destroyed his taste for mathematics. For two years 
his printed copy of the Mecanique, fresh from the press, — the 
work of a quarter of a century, — lay unopened on his desk. 
Through Lavoisier he became interested in chemistry, which 
he found "as easy as algebra." The disastrous crisis of the 


French Revolution aroused him again to activity. About this 
time the young and accomplished daughter of the astronomer 
Lemonnier took compassion on the sad, lonely Lagrange, and 
insisted upon marrying him. Her devotion to him constituted 
the one tie to life which at the approach of death he found it 
hard to break. 

He was made one of the commissioners to establish weights 
and measures having units founded on nature. Lagrange 
strongly favoured the decimal subdivision, the general idea of 
which was obtained from a work of Thomas Williams, London, 
1788. Such was the moderation of Lagrange's character, and 
such the universal respect for him, that he was retained as 
president of the commission on weights and measures even 
after it had been purified by the Jacobins by striking out the 
names of Lavoisier, Laplace, and others. Lagrange took alarm 
at the fate of Lavoisier, and planned to return to Berlin, but 
at the establishment of the Ecole Normale in 1795 in Paris, 
he was induced to accept a professorship. Scarcely had he 
time to elucidate the foundations of arithmetic and algebra 
to young pupils, when the school was closed. His additions 
to the algebra of Euler were prepared at this time. In 1797 
the Ecole Polytechnique was founded, with Lagrange as one 
of the professors. The earliest triumph of this institution was 
the restoration of Lagrange to analysis. His mathematical 
activity burst out anew. He brought forth the Theorie 
des fonctions analytiques (1797), Legons sur le calcul des 
fonctions, a treatise on the same lines as the preceding (1801), 
and the Resolution des equations numeriques (1798). In 1810 


he began a thorough revision of his Mecanique analytique, but 
he died before its completion. 

The Theorie des fonctions, the germ of which is found in 
a memoir of his of 1772, aimed to place the principles of the 
calculus upon a sound foundation by relieving the mind of the 
difficult conception of a limit or infinitesimal. John Landen's 
residual calculus, professing a similar object, was unknown 
to him. Lagrange attempted to prove Taylor's theorem (the 
power of which he was the first to point out) by simple algebra, 
and then to develop the entire calculus from that theorem. 
The principles of the calculus were in his day involved in 
philosophic difficulties of a serious nature. The infinitesimals 
of Leibniz had no satisfactory metaphysical basis. In the 
differential calculus of Euler they were treated as absolute 
zeros. In Newton's limiting ratio, the magnitudes of which 
it is the ratio cannot be found, for at the moment when 
they should be caught and equated, there is neither arc nor 
chord. The chord and arc were not taken by Newton as equal 
before vanishing, nor after vanishing, but when they vanish. 
"That method," said Lagrange, "has the great inconvenience 
of considering quantities in the state in which they cease, so 
to speak, to be quantities; for though we can always well 
conceive the ratios of two quantities, as long as they remain 
finite, that ratio offers to the mind no clear and precise idea, 
as soon as its terms become both nothing at the same time." 
D'Alembert's method of limits was much the same as the 
method of prime and ultimate ratios. D'Alembert taught 
that a variable actually reached its limit. When Lagrange 


endeavoured to free the calculus of its metaphysical difficulties, 
by resorting to common algebra, he avoided the whirlpool of 
Charybdis only to suffer wreck against the rocks of Scylla. 
The algebra of his day, as handed down to him by Euler, 
was founded on a false view of infinity No correct theory of 
infinite series had then been established. Lagrange proposed 
to define the differential coefficient of f(x) with respect to x 
as the coefficient of h in the expansion of f(x + h) by Taylor's 
theorem, and thus to avoid all reference to limits. But he used 
infinite series without ascertaining that they were convergent, 
and his proof that f(x + h) can always be expanded in a 
series of ascending powers of h, labours under serious defects. 
Though Lagrange's method of developing the calculus was at 
first greatly applauded, its defects were fatal, and to-day his 
"method of derivatives," as it was called, has been generally 
abandoned. He introduced a notation of his own, but it 
was inconvenient, and was abandoned by him in the second 
edition of his Mecanique, in which he used infinitesimals. The 
primary object of the Theorie des fonctions was not attained, 
but its secondary results were far-reaching. It was a purely 
abstract mode of regarding functions, apart from geometrical 
or mechanical considerations. In the further development 
of higher analysis a function became the leading idea, and 
Lagrange's work may be regarded as the starting-point of 
the theory of functions as developed by Cauchy, Riemann, 
Weierstrass, and others. 

In the treatment of infinite series Lagrange displayed in his 
earlier writings that laxity common to all mathematicians of 


his time, excepting Nicolaus Bernoulli II. and D'Alembert. 
But his later articles mark the beginning of a period of 
greater rigour. Thus, in the Calcul de fonctions he gives 
his theorem on the limits of Taylor's theorem. Lagrange's 
mathematical researches extended to subjects which have not 
been mentioned here — such as probabilities, finite differences, 
ascending continued fractions, elliptic integrals. Everywhere 
his wonderful powers of generalisation and abstraction are 
made manifest. In that respect he stood without a peer, but 
his great contemporary, Laplace, surpassed him in practical 
sagacity. Lagrange was content to leave the application of 
his general results to others, and some of the most important 
researches of Laplace (particularly those on the velocity of 
sound and on the secular acceleration of the moon) are 
implicitly contained in Lagrange's works. 

Lagrange was an extremely modest man, eager to avoid 
controversy, and even timid in conversation. He spoke in tones 
of doubt, and his first words generally were, "Je ne sais pas." 
He would never allow his portrait to be taken, and the only 
ones that were secured were sketched without his knowledge 
by persons attending the meetings of the Institute. 

Pierre Simon Laplace (1749-1827) was born at Beau- 
mont-en-Auge in Normandy. Very little is known of his early 
life. When at the height of his fame he was loath to speak of 
his boyhood, spent in poverty. His father was a small farmer. 
Some rich neighbours who recognised the boy's talent assisted 
him in securing an education. As an extern he attended the 
military school in Beaumont, where at an early age he became 


teacher of mathematics. At eighteen he went to Paris, armed 
with letters of recommendation to D'Alembert, who was then 
at the height of his fame. The letters remained unnoticed, but 
young Laplace, undaunted, wrote the great geometer a letter 
on the principles of mechanics, which brought the following 
enthusiastic response: "You needed no introduction; you have 
recommended yourself; my support is your due." D'Alembert 
secured him a position at the Ecole Militaire of Paris as 
professor of mathematics. His future was now assured, and 
he entered upon those profound researches which brought 
him the title of "the Newton of France." With wonderful 
mastery of analysis, Laplace attacked the pending problems in 
the application of the law of gravitation to celestial motions. 
During the succeeding fifteen years appeared most of his 
original contributions to astronomy. His career was one 
of almost uninterrupted prosperity. In 1784 he succeeded 
Bezout as examiner to the royal artillery, and the following 
year he became member of the Academy of Sciences. He 
was made president of the Bureau of Longitude; he aided 
in the introduction of the decimal system, and taught, with 
Lagrange, mathematics in the Ecole Normale. When, during 
the Revolution, there arose a cry for the reform of everything, 
even of the calendar, Laplace suggested the adoption of an 
era beginning with the year 1250, when, according to his 
calculation, the major axis of the earth's orbit had been 
perpendicular to the equinoctial line. The year was to begin 
with the vernal equinox, and the zero meridian was to be 
located east of Paris by 185.30 degrees of the centesimal 


division of the quadrant, for by this meridian the beginning 
of his proposed era fell at midnight. But the revolutionists 
rejected this scheme, and made the start of the new era coincide 
with the beginning of the glorious French Republic. [50] 

Laplace was justly admired throughout Europe as a most 
sagacious and profound scientist, but, unhappily for his 
reputation, he strove not only after greatness in science, 
but also after political honours. The political career of this 
eminent scientist was stained by servility and suppleness. 
After the 18th of Brumaire, the day when Napoleon was made 
emperor, Laplace's ardour for republican principles suddenly 
gave way to a great devotion to the emperor. Napoleon 
rewarded this devotion by giving him the post of minister of 
the interior, but dismissed him after six months for incapacity. 
Said Napoleon, "Laplace ne saisissait aucune question sous 
son veritable point de vue; il cherchait des subtilites partout, 
n'avait que des idees problematiques, et portait enfin l'esprit 
des infiniment petits jusque dans 1' administration." Desirous 
to retain his allegiance, Napoleon elevated him to the Senate 
and bestowed various other honours upon him. Nevertheless, 
he cheerfully gave his voice in 1814 to the dethronement of his 
patron and hastened to tender his services to the Bourbons, 
thereby earning the title of marquis. This pettiness of his 
character is seen in his writings. The first edition of the 
Systeme du monde was dedicated to the Council of Five 
Hundred. To the third volume of the Mecanique Celeste is 
prefixed a note that of all the truths contained in the book, 
that most precious to the author was the declaration he thus 


made of gratitude and devotion to the peace-maker of Europe. 
After this outburst of affection, we are surprised to find in 
the editions of the Theorie analytique des probability's, which 
appeared after the Restoration, that the original dedication 
to the emperor is suppressed. 

Though supple and servile in politics, it must be said 
that in religion and science Laplace never misrepresented or 
concealed his own convictions however distasteful they might 
be to others. In mathematics and astronomy his genius 
shines with a lustre excelled by few. Three great works did 
he give to the scientific world, — the Mecanique Celeste, the 
Exposition du systeme du monde, and the Theorie analytique 
des probabilites. Besides these he contributed important 
memoirs to the French Academy. 

We first pass in brief review his astronomical researches. 
In 1773 he brought out a paper in which he proved that the 
mean motions or mean distances of planets are invariable 
or merely subject to small periodic changes. This was the 
first and most important step in establishing the stability of 
the solar system. [51] To Newton and also to Euler it had 
seemed doubtful whether forces so numerous, so variable in 
position, so different in intensity, as those in the solar system, 
could be capable of maintaining permanently a condition of 
equilibrium. Newton was of the opinion that a powerful hand 
must intervene from time to time to repair the derangements 
occasioned by the mutual action of the different bodies. This 
paper was the beginning of a series of profound researches 
by Lagrange and Laplace on the limits of variation of the 


various elements of planetary orbits, in which the two great 
mathematicians alternately surpassed and supplemented each 
other. Laplace's first paper really grew out of researches on 
the theory of Jupiter and Saturn. The behaviour of these 
planets had been studied by Euler and Lagrange without 
receiving satisfactory explanation. Observation revealed the 
existence of a steady acceleration of the mean motions of our 
moon and of Jupiter and an equally strange diminution of the 
mean motion of Saturn. It looked as though Saturn might 
eventually leave the planetary system, while Jupiter would 
fall into the sun, and the moon upon the earth. Laplace finally 
succeeded in showing, in a paper of 1784-1786, that these 
variations (called the "great inequality" ) belonged to the class 
of ordinary periodic perturbations, depending upon the law 
of attraction. The cause of so influential a perturbation was 
found in the commensurability of the mean motion of the two 

In the study of the Jovian system, Laplace was enabled to 
determine the masses of the moons. He also discovered certain 
very remarkable, simple relations between the movements of 
those bodies, known as "Laws of Laplace." His theory of these 
bodies was completed in papers of 1788 and 1789. These, 
as well as the other papers here mentioned, were published 
in the Memoires presentes par divers savans. The year 1787 
was made memorable by Laplace's announcement that the 
lunar acceleration depended upon the secular changes in the 
eccentricity of the earth's orbit. This removed all doubt then 
existing as to the stability of the solar system. The universal 


validity of the law of gravitation to explain all motion in the 
solar system was established. That system, as then known, 
was at last found to be a complete machine. 

In 1796 Laplace published his Exposition du systeme du 
monde, a non-mathematical popular treatise on astronomy, 
ending with a sketch of the history of the science. In this 
work he enunciates for the first time his celebrated nebular 
hypothesis. A similar theory had been previously proposed 
by Kant in 1755, and by Swedenborg; but Laplace does not 
appear to have been aware of this. 

Laplace conceived the idea of writing a work which should 
contain a complete analytical solution of the mechanical prob- 
lem presented by the solar system, without deriving from 
observation any but indispensable data. The result was the 
Mecanique Celeste, which is a systematic presentation em- 
bracing all the discoveries of Newton, Clairaut, D'Alembert, 
Euler, Lagrange, and of Laplace himself, on celestial mechan- 
ics. The first and second volumes of this work were published 
in 1799; the third appeared in 1802, the fourth in 1805. Of 
the fifth volume, Books XL and XII. were published in 1823; 
Books XIII. , XIV., XV. in 1824, and Book XVI. in 1825. The 
first two volumes contain the general theory of the motions and 
figure of celestial bodies. The third and fourth volumes give 
special theories of celestial motions, — treating particularly of 
motions of comets, of our moon, and of other satellites. The 
fifth volume opens with a brief history of celestial mechanics, 
and then gives in appendices the results of the author's later 
researches. The Mecanique Celeste was such a master-piece, 


and so complete, that Laplace's successors have been able to 
add comparatively little. The general part of the work was 
translated into German by Joh. Karl Burkhardt, and appeared 
in Berlin, 1800-1802. Nathaniel Bowditch brought out an 
edition in English, with an extensive commentary, in Boston, 
1829-1839. The Mecanique Celeste is not easy reading. The 
difficulties lie, as a rule, not so much in the subject itself as 
in the want of verbal explanation. A complicated chain of 
reasoning receives often no explanation whatever. Biot, who 
assisted Laplace in revising the work for the press, tells that he 
once asked Laplace some explanation of a passage in the book 
which had been written not long before, and that Laplace 
spent an hour endeavouring to recover the reasoning which 
had been carelessly suppressed with the remark, "II est facile 
de voir." Notwithstanding the important researches in the 
work, which are due to Laplace himself, it naturally contains 
a great deal that is drawn from his predecessors. It is, in fact, 
the organised result of a century of patient toil. But Laplace 
frequently neglects to properly acknowledge the source from 
which he draws, and lets the reader infer that theorems and 
formulae due to a predecessor are really his own. 

We are told that when Laplace presented Napoleon with a 
copy of the Mecanique Celeste, the latter made the remark, 
"M. Laplace, they tell me you have written this large book on 
the system of the universe, and have never even mentioned 
its Creator." Laplace is said to have replied bluntly, "Je 
n'avais pas besoin de cette hypothese-la." This assertion, 
taken literally, is impious, but may it not have been intended 


to convey a meaning somewhat different from its literal one? 
Newton was not able to explain by his law of gravitation 
all questions arising in the mechanics of the heavens. Thus, 
being unable to show that the solar system was stable, and 
suspecting in fact that it was unstable, Newton expressed the 
opinion that the special intervention, from time to time, of a 
powerful hand was necessary to preserve order. Now Laplace 
was able to prove by the law of gravitation that the solar 
system is stable, and in that sense may be said to have felt no 
necessity for reference to the Almighty. 

We now proceed to researches which belong more properly 
to pure mathematics. Of these the most conspicuous are on 
the theory of probability. Laplace has done more towards 
advancing this subject than any one other investigator. He 
published a series of papers, the main results of which were 
collected in his Theorie analytique des probabilites, 1812. The 
third edition (1820) consists of an introduction and two books. 
The introduction was published separately under the title, 
Essai philosophique sur les probabilites, and is an admirable 
and masterly exposition without the aid of analytical formulae 
of the principles and applications of the science. The first book 
contains the theory of generating functions, which are applied, 
in the second book, to the theory of probability. Laplace gives 
in his work on probability his method of approximation to the 
values of definite integrals. The solution of linear differential 
equations was reduced by him to definite integrals. One of 
the most important parts of the work is the application of 
probability to the method of least squares, which is shown to 


give the most probable as well as the most convenient results. 

The first printed statement of the principle of least squares 
was made in 1806 by Legendre, without demonstration. Gauss 
had used it still earlier, but did not publish it until 1809. The 
first deduction of the law of probability of error that appeared 
in print was given in 1808 by Robert Adrain in the Analyst, a 
journal published by himself in Philadelphia. [2] Proofs of this 
law have since been given by Gauss, Ivory, Herschel, Hagen, 
and others; but all proofs contain some point of difficulty 
Laplace's proof is perhaps the most satisfactory. 

Laplace's work on probability is very difficult reading, 
particularly the part on the method of least squares. The 
analytical processes are by no means clearly established or 
free from error. "No one was more sure of giving the result of 
analytical processes correctly, and no one ever took so little 
care to point out the various small considerations on which 
correctness depends" (De Morgan). 

Of Laplace's papers on the attraction of ellipsoids, the 
most important is the one published in 1785, and to a great 
extent reprinted in the third volume of the Mecanique Celeste. 
It gives an exhaustive treatment of the general problem of 
attraction of any ellipsoid upon a particle situated outside 
or upon its surface. Spherical harmonics, or the so-called 
"Laplace's coefficients," constitute a powerful analytic engine 
in the theory of attraction, in electricity, and magnetism. The 
theory of spherical harmonics for two dimensions had been 
previously given by Legendre. Laplace failed to make due 
acknowledgment of this, and there existed, in consequence, 


between the two great men, "a feeling more than coldness." 
The potential function, V, is much used by Laplace, and 

is shown by him to satisfy the partial differential equation 

d 2 V d 2 V d 2 V 

-— t> + — -s- + -7—9- = 0. This is known as Laplace's equation, 

ox A oy z oz z 

and was first given by him in the more complicated form which 
it assumes in polar co-ordinates. The notion of potential was, 
however, not introduced into analysis by Laplace. The honour 
of that achievement belongs to Lagrange. [49] 

Among the minor discoveries of Laplace are his method of 
solving equations of the second, third, and fourth degrees, 
his memoir on singular solutions of differential equations, 
his researches in finite differences and in determinants, the 
establishment of the expansion theorem in determinants which 
had been previously given by Vandermonde for a special case, 
the determination of the complete integral of the linear 
differential equation of the second order. In the Mecanique 
Celeste he made a generalisation of Lagrange's theorem on 
the development of functions in series known as Laplace's 

Laplace's investigations in physics were quite extensive. 
We mention here his correction of Newton's formula on 
the velocity of sound in gases by taking into account the 
changes of elasticity due to the heat of compression and 
cold of rarefaction; his researches on the theory of tides; 
his mathematical theory of capillarity; his explanation of 
astronomical refraction; his formulae for measuring heights by 
the barometer. 


Laplace's writings stand out in bold contrast to those of 
Lagrange in their lack of elegance and symmetry. Laplace 
looked upon mathematics as the tool for the solution of 
physical problems. The true result being once reached, he 
spent little time in explaining the various steps of his analysis, 
or in polishing his work. The last years of his life were spent 
mostly at Arcueil in peaceful retirement on a country-place, 
where he pursued his studies with his usual vigour until his 
death. He was a great admirer of Euler, and would often say, 
"Lisez Euler, lisez Euler, c'est notre maitre a tous." 

Abnit-Theophile Vandermonde (1735-1796) studied 
music during his youth in Paris and advocated the theory that 
all art rested upon one general law, through which any one 
could become a composer with the aid of mathematics. He was 
the first to give a connected and logical exposition of the theory 
of determinants, and may, therefore, almost be regarded as 
the founder of that theory. He and Lagrange originated the 
method of combinations in solving equations. [20] 

Adrien Marie Legendre (1752-1833) was educated at 
the College Mazarin in Paris, where he began the study of 
mathematics under Abbe Marie. His mathematical genius 
secured for him the position of professor of mathematics at the 
military school of Paris. While there he prepared an essay on 
the curve described by projectiles thrown into resisting media 
(ballistic curve), which captured a prize offered by the Royal 
Academy of Berlin. In 1780 he resigned his position in order 
to reserve more time for the study of higher mathematics. He 
was then made member of several public commissions. In 


1795 he was elected professor at the Normal School and later 
was appointed to some minor government positions. Owing to 
his timidity and to Laplace's unfriendliness toward him, but 
few important public offices commensurate with his ability 
were tendered to him. 

As an analyst, second only to Laplace and Lagrange, 
Legendre enriched mathematics by important contributions, 
mainly on elliptic integrals, theory of numbers, attraction 
of ellipsoids, and least squares. The most important of 
Legendre's works is his Fonctions elliptiques, issued in two 
volumes in 1825 and 1826. He took up the subject where 
Euler, Landen, and Lagrange had left it, and for forty years 
was the only one to cultivate this new branch of analysis, 
until at last Jacobi and Abel stepped in with admirable 
new discoveries. [52] Legendre imparted to the subject that 
connection and arrangement which belongs to an independent 
science. Starting with an integral depending upon the square 
root of a polynomial of the fourth degree in x, he showed 
that such integrals can be brought back to three canonical 
forms, designated by F ((/)), E ((/>), and n(^>), the radical 
being expressed in the form A(<^>) = yl — fc 2 sin 2 (/>. He also 
undertook the prodigious task of calculating tables of arcs of 
the ellipse for different degrees of amplitude and eccentricity, 
which supply the means of integrating a large number of 

An earlier publication which contained part of his researches 
on elliptic functions was his Calcul integral in three volumes 
(1811, 1816, 1817), in which he treats also at length of the 


two classes of definite integrals named by him Eulerian. He 
tabulated the values of log T(p) for values of p between 1 and 2. 

One of the earliest subjects of research was the attraction 
of spheroids, which suggested to Legendre the function P n , 
named after him. His memoir was presented to the Academy 
of Sciences in 1783. The researches of Maclaurin and Lagrange 
suppose the point attracted by a spheroid to be at the surface 
or within the spheroid, but Legendre showed that in order to 
determine the attraction of a spheroid on any external point 
it suffices to cause the surface of another spheroid described 
upon the same foci to pass through that point. Other memoirs 
on ellipsoids appeared later. 

The two household gods to which Legendre sacrificed with 
ever-renewed pleasure in the silence of his closet were the 
elliptic functions and the theory of numbers. His researches 
on the latter subject, together with the numerous scattered 
fragments on the theory of numbers due to his predecessors 
in this line, were arranged as far as possible into a systematic 
whole, and published in two large quarto volumes, entitled 
Theorie des nombres, 1830. Before the publication of this 
work Legendre had issued at divers times preliminary articles. 
Its crowning pinnacle is the theorem of quadratic reciprocity, 
previously indistinctly given by Euler without proof, but 
for the first time clearly enunciated and partly proved by 
Legendre. [48] 

While acting as one of the commissioners to connect 
Greenwich and Paris geodetically, Legendre calculated all the 
triangles in France. This furnished the occasion of establishing 


formulae and theorems on geodesies, on the treatment of the 
spherical triangle as if it were a plane triangle, by applying 
certain corrections to the angles, and on the method of 
least squares, published for the first time by him without 
demonstration in 1806. 

Legendre wrote an Elements de Geometrie, 1794, which 
enjoyed great popularity, being generally adopted on the 
Continent and in the United States as a substitute for 
Euclid. This great modern rival of Euclid passed through 
numerous editions; the later ones containing the elements of 
trigonometry and a proof of the irrationality of n and n 2 . Much 
attention was given by Legendre to the subject of parallel lines. 
In the earlier editions of the Elements, he made direct appeal 
to the senses for the correctness of the "parallel-axiom." He 
then attempted to demonstrate that "axiom," but his proofs 
did not satisfy even himself. In Vol. XII. of the Memoirs of the 
Institute is a paper by Legendre, containing his last attempt 
at a solution of the problem. Assuming space to be infinite, 
he proved satisfactorily that it is impossible for the sum of the 
three angles of a triangle to exceed two right angles; and that 
if there be any triangle the sum of whose angles is two right 
angles, then the same must be true of all triangles. But in 
the next step, to show that this sum cannot be less than two 
right angles, his demonstration necessarily failed. If it could 
be granted that the sum of the three angles is always equal to 
two right angles, then the theory of parallels could be strictly 

Joseph Fourier (1768-1830) was born at Auxerre, in 


central France. He became an orphan in his eighth year. 
Through the influence of friends he was admitted into the 
military school in his native place, then conducted by the 
Benedictines of the Convent of St. Mark. He there prosecuted 
his studies, particularly mathematics, with surprising success. 
He wished to enter the artillery, but, being of low birth (the 
son of a tailor), his application was answered thus: "Fourier, 
not being noble, could not enter the artillery, although 
he were a second Newton." [53] He was soon appointed 
to the mathematical chair in the military school. At the 
age of twenty-one he went to Paris to read before the 
Academy of Sciences a memoir on the resolution of numerical 
equations, which was an improvement on Newton's method of 
approximation. This investigation of his early youth he never 
lost sight of. He lectured upon it in the Polytechnic School; 
he developed it on the banks of the Nile; it constituted a part 
of a work entitled Analyse des equationes determines (1831), 
which was in press when death overtook him. This work 
contained "Fourier's theorem" on the number of real roots 
between two chosen limits. Budan had published this result 
as early as 1807, but there is evidence to show that Fourier 
had established it before Budan's publication. These brilliant 
results were eclipsed by the theorem of Sturm, published 
in 1835. 

Fourier took a prominent part at his home in promoting 
the Revolution. Under the French Revolution the arts and 
sciences seemed for a time to flourish. The reformation 
of the weights and measures was planned with grandeur of 


conception. The Normal School was created in 1795, of which 
Fourier became at first pupil, then lecturer. His brilliant 
success secured him a chair in the Polytechnic School, the 
duties of which he afterwards quitted, along with Monge and 
Berthollet, to accompany Napoleon on his campaign to Egypt. 
Napoleon founded the Institute of Egypt, of which Fourier 
became secretary. In Egypt he engaged not only in scientific 
work, but discharged important political functions. After 
his return to France he held for fourteen years the prefecture 
of Grenoble. During this period he carried on his elaborate 
investigations on the propagation of heat in solid bodies, 
published in 1822 in his work entitled La Theorie Analytique 
de la Chaleur. This work marks an epoch in the history 
of mathematical physics. "Fourier's series" constitutes its 
gem. By this research a long controversy was brought to a 
close, and the fact established that any arbitrary function 
can be represented by a trigonometric series. The first 
announcement of this great discovery was made by Fourier in 
1807, before the French Academy. The trigonometric series 

Y^ (a n sin nx + b n cos nx) represents the function <f>(x) for 
n=0 1 f n 

every value of x, if the coefficients a n = — / 4>(x) sinnxdx, 

K J-n 

and b n be equal to a similar integral. The weak point in 
Fourier's analysis lies in his failure to prove generally that 
the trigonometric series actually converges to the value of the 
function. In 1827 Fourier succeeded Laplace as president of 
the council of the Polytechnic School. 

Before proceeding to the origin of modern geometry we 


shall speak briefly of the introduction of higher analysis into 
Great Britain. This took place during the first quarter of this 
century. The British began to deplore the very small progress 
that science was making in England as compared with its 
racing progress on the Continent. In 1813 the "Analytical 
Society" was formed at Cambridge. This was a small 
club established by George Peacock, John Herschel, Charles 
Babbage, and a few other Cambridge students, to promote, as 
it was humorously expressed, the principles of pure "D-ism," 
that is, the Leibnizian notation in the calculus against those 

of "dot-age," or of the Newtonian notation. This struggle 

ended in the introduction into Cambridge of the notation — , 

to the exclusion of the fluxional notation y. This was a great 

step in advance, not on account of any great superiority of 

the Leibnizian over the Newtonian notation, but because the 

adoption of the former opened up to English students the vast 

storehouses of continental discoveries. Sir William Thomson, 

Tait, and some other modern writers find it frequently 

convenient to use both notations. Herschel, Peacock, and 

Babbage translated, in 1816, from the French, Lacroix's 

treatise on the differential and integral calculus, and added in 

1820 two volumes of examples. Lacroix's was one of the best 

and most extensive works on the calculus of that time. Of the 

three founders of the "Analytical Society," Peacock afterwards 

did most work in pure mathematics. Babbage became famous 

for his invention of a calculating engine superior to Pascal's. 

It was never finished, owing to a misunderstanding with the 

government, and a consequent failure to secure funds. John 


Herschel, the eminent astronomer, displayed his mastery over 
higher analysis in memoirs communicated to the Royal Society 
on new applications of mathematical analysis, and in articles 
contributed to cyclopaedias on light, on meteorology, and on 
the history of mathematics. 

George Peacock (1791-1858) was educated at Trinity 
College, Cambridge, became Lowndean professor there, and 
later, dean of Ely. His chief publications are his Algebra, 1830 
and 1842, and his Report on Recent Progress in Analysis, which 
was the first of several valuable summaries of scientific progress 
printed in the volumes of the British Association. He was one 
of the first to study seriously the fundamental principles of 
algebra, and to fully recognise its purely symbolic character. 
He advances, though somewhat imperfectly, the "principle of 
the permanence of equivalent forms." It assumes that the rules 
applying to the symbols of arithmetical algebra apply also 
in symbolical algebra. About this time D. F. Gregory wrote 
a paper "on the real nature of symbolical algebra," which 
brought out clearly the commutative and distributive laws. 
These laws had been noticed years before by the inventors 
of symbolic methods in the calculus. It was Servois who 
introduced the names commutative and distributive in 1813. 
Peacock's investigations on the foundation of algebra were 
considerably advanced by De Morgan and Hankel. 

James Ivory (1765-1842) was a Scotch mathematician who 
for twelve years, beginning in 1804, held the mathematical 
chair in the Royal Military College at Marlow (now at 
Sandhurst). He was essentially a self-trained mathematician, 


and almost the only one in Great Britain previous to the 
organisation of the Analytical Society who was well versed 
in continental mathematics. Of importance is his memoir 
(Phil. Trans., 1809) in which the problem of the attraction 
of a homogeneous ellipsoid upon an external point is reduced 
to the simpler problem of the attraction of a related ellipsoid 
upon a corresponding point interior to it. This is known as 
"Ivory's theorem." He criticised with undue severity Laplace's 
solution of the method of least squares, and gave three proofs 
of the principle without recourse to probability; but they are 
far from being satisfactory. 

The Origin of Modern Geometry. 

By the researches of Descartes and the invention of the 
calculus, the analytical treatment of geometry was brought 
into great prominence for over a century. Notwithstanding 
the efforts to revive synthetic methods made by Desargues, 
Pascal, De Lahire, Newton, and Maclaurin, the analytical 
method retained almost undisputed supremacy. It was 
reserved for the genius of Monge to bring synthetic geometry 
in the foreground, and to open up new avenues of progress. 
His Geometrie descriptive marks the beginning of a wonderful 
development of modern geometry. 

Of the two leading problems of descriptive geometry, the 
one — to represent by drawings geometrical magnitudes — was 
brought to a high degree of perfection before the time of 
Monge; the other — to solve problems on figures in space by 


constructions in a plane — had received considerable attention 
before his time. His most noteworthy predecessor in de- 
scriptive geometry was the Frenchman Frezier (1682-1773). 
But it remained for Monge to create descriptive geometry 
as a distinct branch of science by imparting to it geometric 
generality and elegance. All problems previously treated in 
a special and uncertain manner were referred back to a few 
general principles. He introduced the line of intersection of 
the horizontal and the vertical plane as the axis of projection. 
By revolving one plane into the other around this axis or 
ground-line, many advantages were gained. [54] 

Gaspard Monge (1746-1818) was born at Beaune. The 
construction of a plan of his native town brought the boy 
under the notice of a colonel of engineers, who procured for 
him an appointment in the college of engineers at Mezieres. 
Being of low birth, he could not receive a commission in 
the army, but he was permitted to enter the annex of the 
school, where surveying and drawing were taught. Observing 
that all the operations connected with the construction of 
plans of fortification were conducted by long arithmetical 
processes, he substituted a geometrical method, which the 
commandant at first refused even to look at, so short was the 
time in which it could be practised; when once examined, it 
was received with avidity. Monge developed these methods 
further and thus created his descriptive geometry. Owing to 
the rivalry between the French military schools of that time, 
he was not permitted to divulge his new methods to any one 
outside of this institution. In 1768 he was made professor of 


mathematics at Mezieres. In 1780, when conversing with two 
of his pupils, S. F. Lacroix and Gayvernon in Paris, he was 
obliged to say, "All that I have here done by calculation, I 
could have done with the ruler and compass, but I am not 
allowed to reveal these secrets to you." But Lacroix set himself 
to examine what the secret could be, discovered the processes, 
and published them in 1795. The method was published by 
Monge himself in the same year, first in the form in which the 
short-hand writers took down his lessons given at the Normal 
School, where he had been elected professor, and then again, 
in revised form, in the Journal des ecoles normales. The next 
edition occurred in 1798-1799. After an ephemeral existence 
of only four months the Normal School was closed in 1795. 
In the same year the Polytechnic School was opened, in the 
establishing of which Monge took active part. He taught 
there descriptive geometry until his departure from France to 
accompany Napoleon on the Egyptian campaign. He was the 
first president of the Institute of Egypt. Monge was a zealous 
partisan of Napoleon and was, for that reason, deprived of all 
his honours by Louis XVIII. This and the destruction of the 
Polytechnic School preyed heavily upon his mind. He did not 
long survive this insult. 

Monge's numerous papers were by no means confined to 
descriptive geometry. His analytical discoveries are hardly 
less remarkable. He introduced into analytic geometry the 
methodic use of the equation of a line. He made important 
contributions to surfaces of the second degree (previously 
studied by Wren and Euler) and discovered between the theory 


of surfaces and the integration of partial differential equations, 
a hidden relation which threw new light upon both subjects. 
He gave the differential of curves of curvature, established a 
general theory of curvature, and applied it to the ellipsoid. 
He found that the validity of solutions was not impaired when 
imaginaries are involved among subsidiary quantities. Monge 
published the following books: Statics, 1786; Applications de 
I'algebre a la geometric, 1805; Application de V analyse a la 
geometric The last two contain most of his miscellaneous 

Monge was an inspiring teacher, and he gathered around 
him a large circle of pupils, among which were Dupin, Servois, 
Brianchon, Hachette, Biot, and Poncelet. 

Charles Dupin (1784-1873), for many years professor of 
mechanics in the Conservatoire des Arts et Metiers in Paris, 
published in 1813 an important work on Developpements de 
geometrie, in which is introduced the conception of conjugate 
tangents of a point of a surface, and of the indicatrix. [53] 
It contains also the theorem known as "Dupin's theorem." 
Surfaces of the second degree and descriptive geometry were 
successfully studied by Jean Nicolas Pierre Hachette (1769- 
1834), who became professor of descriptive geometry at the 
Polytechnic School after the departure of Monge for Rome 
and Egypt. In 1822 he published his Traite de geometrie 

Descriptive geometry, which arose, as we have seen, in 
technical schools in France, was transferred to Germany at 
the foundation of technical schools there. G. Schreiber, 


professor in Karlsruhe, was the first to spread Monge's 
geometry in Germany by the publication of a work thereon 
in 1828-1829. [54] In the United States descriptive geometry 
was introduced in 1816 at the Military Academy in West Point 
by Claude Crozet, once a pupil at the Polytechnic School in 
Paris. Crozet wrote the first English work on the subject. [2] 

Lazare Nicholas Marguerite Carnot (1753-1823) was 
born at Nolay in Burgundy, and educated in his native 
province. He entered the army, but continued his math- 
ematical studies, and wrote in 1784 a work on machines, 
containing the earliest proof that kinetic energy is lost in 
collisions of bodies. With the advent of the Revolution he 
threw himself into politics, and when coalesced Europe, in 
1793, launched against France a million soldiers, the gigantic 
task of organising fourteen armies to meet the enemy was 
achieved by him. He was banished in 1796 for opposing 
Napoleon's coup d'etat. The refugee went to Geneva, where 
he issued, in 1797, a work still frequently quoted, entitled, 
Reflexions sur la Metaphysique du Calcul Infinitesimal. He 
declared himself as an "irreconcilable enemy of kings." After 
the Russian campaign he offered to fight for France, though 
not for the empire. On the restoration he was exiled. He died 
in Magdeburg. His Geometrie de position, 1803, and his Essay 
on Transversals, 1806, are important contributions to modern 
geometry. While Monge revelled mainly in three-dimensional 
geometry, Carnot confined himself to that of two. By his 
effort to explain the meaning of the negative sign in geometry 
he established a "geometry of position," which, however, 


is different from the "Geometrie der Lage" of to-day. He 
invented a class of general theorems on projective properties 
of figures, which have since been pushed to great extent by 
Poncelet, Chasles, and others. 

Jean Victor Poncelet (1788-1867), a native of Metz, 
took part in the Russian campaign, was abandoned as dead 
on the bloody field of Krasnoi, and taken prisoner to Saratoff. 
Deprived there of all books, and reduced to the remembrance 
of what he had learned at the Lyceum at Metz and the 
Polytechnic School, where he had studied with predilection 
the works of Monge, Carnot, and Brianchon, he began to 
study mathematics from its elements. He entered upon 
original researches which afterwards made him illustrious. 
While in prison he did for mathematics what Bunyan did for 
literature, — produced a much-read work, which has remained 
of great value down to the present time. He returned to 
France in 1814, and in 1822 published the work in question, 
entitled, Traite des Proprietes projectives des figures. In it he 
investigated the properties of figures which remain unaltered 
by projection of the figures. The projection is not effected 
here by parallel rays of prescribed direction, as with Monge, 
but by central projection. Thus perspective projection, used 
before him by Desargues, Pascal, Newton, and Lambert, was 
elevated by him into a fruitful geometric method. In the 
same way he elaborated some ideas of De Lahire, Servois, and 
Gergonne into a regular method — the method of "reciprocal 
polars." To him we owe the Law of Duality as a consequence 
of reciprocal polars. As an independent principle it is due to 


Gergonne. Poncelet wrote much on applied mechanics. In 
1838 the Faculty of Sciences was enlarged by his election to 
the chair of mechanics. 

While in France the school of Monge was creating modern 
geometry, efforts were made in England to revive Greek 
geometry by Robert Simson (1687-1768) and Matthew 
Stewart (1717-1785). Stewart was a pupil of Simson 
and Maclaurin, and succeeded the latter in the chair at 
Edinburgh. During the eighteenth century he and Maclaurin 
were the only prominent mathematicians in Great Britain. 
His genius was ill-directed by the fashion then prevalent 
in England to ignore higher analysis. In his Four Tracts, 
Physical and Mathematical, 1761, he applied geometry to 
the solution of difficult astronomical problems, which on the 
Continent were approached analytically with greater success. 
He published, in 1746, General Theorems, and in 1763, his 
Propositiones geometric^ more veterum demonstrate. The 
former work contains sixty-nine theorems, of which only 
five are accompanied by demonstrations. It gives many 
interesting new results on the circle and the straight line. 
Stewart extended some theorems on transversals due to 
Giovanni Ceva (1648-1737), an Italian, who published in 1678 
at Mediolani a work containing the theorem now known by 
his name. 


Never more zealously and successfully has mathematics 
been cultivated than in this century. Nor has progress, as 
in previous periods, been confined to one or two countries. 
While the French and Swiss, who alone during the preceding 
epoch carried the torch of progress, have continued to develop 
mathematics with great success, from other countries whole 
armies of enthusiastic workers have wheeled into the front 
rank. Germany awoke from her lethargy by bringing forward 
Gauss, Jacobi, Dirichlet, and hosts of more recent men; 
Great Britain produced her De Morgan, Boole, Hamilton, 
besides champions who are still living; Russia entered the 
arena with her Lobatchewsky; Norway with Abel; Italy with 
Cremona; Hungary with her two Bolyais; the United States 
with Benjamin Peirce. 

The productiveness of modern writers has been enormous. 
"It is difficult," says Professor Cayley, [56] "to give an idea of 
the vast extent of modern mathematics. This word 'extent' 
is not the right one: I mean extent crowded with beautiful 
detail, — not an extent of mere uniformity such as an objectless 
plain, but of a tract of beautiful country seen at first in the 
distance, but which will bear to be rambled through and 
studied in every detail of hillside and valley, stream, rock, 
wood, and flower." It is pleasant to the mathematician to 
think that in his, as in no other science, the achievements of 



every age remain possessions forever; new discoveries seldom 
disprove older tenets; seldom is anything lost or wasted. 

If it be asked wherein the utility of some modern extensions 
of mathematics lies, it must be acknowledged that it is at 
present difficult to see how they are ever to become applicable 
to questions of common life or physical science. But our 
inability to do this should not be urged as an argument 
against the pursuit of such studies. In the first place, we know 
neither the day nor the hour when these abstract developments 
will find application in the mechanic arts, in physical science, 
or in other branches of mathematics. For example, the 
whole subject of graphical statics, so useful to the practical 
engineer, was made to rest upon von Staudt's Geometrie 
der Lage; Hamilton's "principle of varying action" has its 
use in astronomy; complex quantities, general integrals, and 
general theorems in integration offer advantages in the study 
of electricity and magnetism. "The utility of such researches," 
says Spottiswoode, [57] "can in no case be discounted, or even 
imagined beforehand. Who, forinstance, would have supposed 
that the calculus of forms or the theory of substitutions would 
have thrown much light upon ordinary equations; or that 
Abelian functions and hyperelliptic transcendents would have 
told us anything about the properties of curves; or that 
the calculus of operations would have helped us in any way 
towards the figure of the earth?" A second reason in favour 
of the pursuit of advanced mathematics, even when there is 
no promise of practical application, is this, that mathematics, 
like poetry and music, deserves cultivation for its own sake. 


The great characteristic of modern mathematics is its 
generalising tendency. Nowadays little weight is given to 
isolated theorems, "except as affording hints of an unsuspected 
new sphere of thought, like meteorites detached from some 
undiscovered planetary orb of speculation." In mathematics, 
as in all true sciences, no subject is considered in itself alone, 
but always as related to, or an outgrowth of, other things. The 
development of the notion of continuity plays a leading part in 
modern research. In geometry the principle of continuity, the 
idea of correspondence, and the theory of projection constitute 
the fundamental modern notions. Continuity asserts itself in 
a most striking way in relation to the circular points at infinity 
in a plane. In algebra the modern idea finds expression in the 
theory of linear transformations and invariants, and in the 
recognition of the value of homogeneity and symmetry. 


The conflict between geometry and analysis which arose 
near the close of the last century and the beginning of the 
present has now come to an end. Neither side has come 
out victorious. The greatest strength is found to lie, not in 
the suppression of either, but in the friendly rivalry between 
the two, and in the stimulating influence of the one upon 
the other. Lagrange prided himself that in his Mecanique 
Analytique he had succeeded in avoiding all figures; but since 
his time mechanics has received much help from geometry. 

Modern synthetic geometry was created by several investi- 


gators about the same time. It seemed to be the outgrowth of 
a desire for general methods which should serve as threads of 
Ariadne to guide the student through the labyrinth of theo- 
rems, corollaries, porisms, and problems. Synthetic geometry 
was first cultivated by Monge, Carnot, and Poncelet in France; 
it then bore rich fruits at the hands of Mobius and Steiner 
in Germany and Switzerland, and was finally developed to 
still higher perfection by Chasles in France, von Staudt in 
Germany, and Cremona in Italy. 

Augustus Ferdinand Mobius (1790-1868) was a native 
of Schulpforta in Prussia. He studied at Gottingen under 
Gauss, also at Leipzig and Halle. In Leipzig he became, in 
1815, privat-docent, the next year extraordinary professor of 
astronomy, and in 1844 ordinary professor. This position he 
held till his death. The most important of his researches are 
on geometry. They appeared in Crelle 's Journal, and in his 
celebrated work entitled Der Barycentrische Calcul, Leipzig, 
1827. As the name indicates, this calculus is based upon 
properties of the centre of gravity. [58] Thus, that the point S 
is the centre of gravity of weights a, 6, c, d placed at the points 
A, B, C, D respectively, is expressed by the equation 

(a + b + c + d)S = aA + bB + cC + dD. 

His calculus is the beginning of a quadruple algebra, and 
contains the germs of Grassmann's marvellous system. In 
designating segments of lines we find throughout this work 
for the first time consistency in the distinction of positive 
and negative by the order of letters AB, BA. Similarly 


for triangles and tetrahedra. The remark that it is always 
possible to give three points A, B, C such weights a, (3, 7 
that any fourth point M in their plane will become a centre 
of mass, led Mobius to a new system of co-ordinates in 
which the position of a point was indicated by an equation, 
and that of a line by co-ordinates. By this algorithm he 
found by algebra many geometric theorems expressing mainly 
invariantal properties, — for example, the theorems on the 
anharmonic relation. Mobius wrote also on statics and 
astronomy. He generalised spherical trigonometry by letting 
the sides or angles of triangles exceed 180°. 

Jacob Steiner (1796-1863), "the greatest geometrician 
since the time of Euclid," was born in Utzendorf in the 
Canton of Bern. He did not learn to write till he was 
fourteen. At eighteen he became a pupil of Pestalozzi. 
Later he studied at Heidelberg and Berlin. When Crelle 
started, in 1826, the celebrated mathematical journal bearing 
his name, Steiner and Abel became leading contributors. 
In 1832 Steiner published his Systematische Entwickelung 
der Abhdngigkeit geometrischer Gestalten von einander, "in 
which is uncovered the organism by which the most diverse 
phenomena (Erscheinungen) in the world of space are united 
to each other." Through the influence of Jacobi and others, 
the chair of geometry was founded for him at Berlin in 1834. 
This position he occupied until his death, which occurred after 
years of bad health. In his Systematische Entwickelung en, 
for the first time, is the principle of duality introduced at 
the outset. This book and von Staudt's lay the foundation 


on which synthetic geometry in its present form rests. Not 
only did he fairly complete the theory of curves and surfaces 
of the second degree, but he made great advances in the 
theory of those of higher degrees. In his hands synthetic 
geometry made prodigious progress. New discoveries followed 
each other so rapidly that he often did not take time to 
record their demonstrations. In an article in Crelle's Journal 
on Allgemeine Eigenschaften Algebraischer Curven he gives 
without proof theorems which were declared by Hesse to be 
"like Fermat's theorems, riddles to the present and future 
generations." Analytical proofs of some of them have been 
given since by others, but Cremona finally proved them all by 
a synthetic method. Steiner discovered synthetically the two 
prominent properties of asurface of the third order; viz. that it 
contains twenty-seven straight lines and a pentahedron which 
has the double points for its vertices and the lines of the Hessian 
of the given surface for its edges. [55] The first property was 
discovered analytically somewhat earlier in England by Cayley 
and Salmon, and the second by Sylvester. Steiner's work on 
this subject was the starting-point of important researches by 
H. Schroter, F. August, L. Cremona, and R. Sturm. Steiner 
made investigations by synthetic methods on maxima and 
minima, and arrived at the solution of problems which at 
that time altogether surpassed the analytic power of the 
calculus of variations. He generalised the hexagrammum 
mysticum and also Malfatti's problem. [59] Malfatti, in 1803, 
proposed the problem, to cut three cylindrical holes out of 
a three-sided prism in such a way that the cylinders and 


the prism have the same altitude and that the volume of 
the cylinders be a maximum. This problem was reduced 
to another, now generally known as Malfatti's problem: to 
inscribe three circles in a triangle that each circle will be 
tangent to two sides of a triangle and to the other two circles. 
Malfatti gave an analytical solution, but Steiner gave without 
proof a construction, remarked that there were thirty-two 
solutions, generalised the problem by replacing the three lines 
by three circles, and solved the analogous problem for three 
dimensions. This general problem was solved analytically by 
C. H. Schellbach (1809-1892) and Cayley, and by Clebsch 
with the aid of the addition theorem of elliptic functions. [60] 

Steiner's researches are confined to synthetic geometry. He 
hated analysis as thoroughly as Lagrange disliked geometry. 
Steiner's Gesammelte Werke were published in Berlin in 1881 
and 1882. 

Michel Chasles (1793-1880) was born at Epernon, entered 
the Polytechnic School of Paris in 1812, engaged afterwards 
in business, which he later gave up that he might devote 
all his time to scientific pursuits. In 1841 he became 
professor of geodesy and mechanics at the Polytechnic School; 
later, "Professeur de Geometrie superieure a la Faculte des 
Sciences de Paris." He was a voluminous writer on geometrical 
subjects. In 1837 he published his admirable Apergu historique 
sur I'origine et le developpement des methodes en geometrie, 
containing a history of geometry and, as an appendix, atreatise 
"sur deux principes generaux de la Science." The Apergu 
historique is still a standard historical work; the appendix 


contains the general theory of Homography (Collineation) 
and of duality (Reciprocity). The name duality is due to 
Joseph Diaz Gergonne (1771-1859). Chasles introduced 
the term anharmonic ratio, corresponding to the German 
Doppelverhaltniss and to Clifford's cross-ratio. Chasles and 
Steiner elaborated independently the modern synthetic or 
projective geometry. Numerous original memoirs of Chasles 
were published later in the Journal de I'Ecole Polytechnique. 
He gave a reduction of cubics, different from Newton's in this, 
that the five curves from which all others can be projected 
are symmetrical with respect to a centre. In 1864 he began 
the publication, in the Comptes rendus, of articles in which he 
solves by his "method of characteristics" and the "principle 
of correspondence" an immense number of problems. He 
determined, for instance, the number of intersections of two 
curves in a plane. The method of characteristics contains 
the basis of enumerative geometry. The application of the 
principle of correspondence was extended by Cayley, A. Brill, 
H. G. Zeuthen, H. A. Schwarz, G. H. Halphen (1844-1889), 
and others. The full value of these principles of Chasles 
was not brought out until the appearance, in 1879, of the 
Kalkiil der Abzahlenden Geometrie by Hermann Schubert of 
Hamburg. This work contains a masterly discussion of the 
problem of enumerative geometry, viz. to determine how many 
geometric figures of given definition satisfy a sufficient number 
of conditions. Schubert extended his enumerative geometry 
to n-dimensional space. [55] 

To Chasles we owe the introduction into projective geom- 


etry of non-pro jective properties of figures by means of the 
infinitely distant imaginary sphero- circle. [61] Remarkable 
is his complete solution, in 1846, by synthetic geometry, of 
the difficult question of the attraction of an ellipsoid on an 
external point. This was accomplished analytically by Poisson 
in 1835. The labours of Chasles and Steiner raised synthetic 
geometry to an honoured and respected position by the side 
of analysis. 

Karl Georg Christian von Staudt (1798-1867) was born 
in Rothenburg on the Tauber, and, at his death, was professor 
in Erlangen. His great works are the Geometrie der Lage, 
Nurnberg, 1847, and his Beitrdge zur Geometrie der Lage, 
1856-1860. The author cut loose from algebraic formulas and 
from metrical relations, particularly the anharmonic ratio of 
Steiner and Chasles, and then created a geometry of position, 
which is a complete science in itself, independent of all 
measurements. He shows that projective properties of figures 
have no dependence whatever on measurements, and can be 
established without any mention of them. In his theory of 
what he calls "Wurfe," he even gives a geometrical definition 
of a number in its relation to geometry as determining the 
position of a point. The Beitrdge contains the first complete 
and general theory of imaginary points, lines, and planes in 
projective geometry. Representation of an imaginary point is 
sought in the combination of an involution with a determinate 
direction, both on the real line through the point. While 
purely projective, von Staudt 's method is intimately related 
to the problem of representing by actual points and lines the 


imaginaries of analytical geometry. This was systematically 
undertaken by C. F. Maximilien Marie, who worked, however, 
on entirely different lines. An independent attempt has been 
made recently (1893) by F. H. Loud of Colorado College. 
Von Staudt's geometry of position was for a long time 
disregarded, mainly, no doubt, because his book is extremely 
condensed. An impulse to the study of this subject was given 
by Culmann, who rests his graphical statics upon the work of 
von Staudt. An interpreter of von Staudt was at last found in 
Theodor Reye of Strassburg, who wrote a Geometrie derLage 
in 1868. 

Synthetic geometry has been studied with much success by 
Luigi Cremona, professor in the University of Rome. In his 
Introduzione ad una teoria geometrica delle curve piane he 
developed by a uniform method many new results and proved 
synthetically all important results reached before that time by 
analysis. His writings have been translated into German by 
M. Curtze, professor at the gymnasium in Thorn. The theory 
of the transformation of curves and of the correspondence of 
points on curves was extended by him to three dimensions. 
Ruled surfaces, surfaces of the second order, space-curves 
of the third order, and the general theory of surfaces have 
received much attention at his hands. 

Karl Culmann, professor at the Polytechnicum in Zurich, 
published an epoch-making work on Die graphische Statik, 
Zurich, 1864, which has rendered graphical statics a great 
rival of analytical statics. Before Culmann, B. E. Cousinery 
had turned his attention to the graphical calculus, but he 


made use of perspective, and not of modern geometry. [62] 
Culmann is the first to undertake to present the graphical 
calculus as a symmetrical whole, holding the same relation to 
the new geometry that analytical mechanics does to higher 
analysis. He makes use of the polar theory of reciprocal 
figures as expressing the relation between the force and 
the funicular polygons. He deduces this relation without 
leaving the plane of the two figures. But if the polygons 
be regarded as projections of lines in space, these lines may 
be treated as reciprocal elements of a "Nullsystem." This 
was done by Clerk Maxwell in 1864, and elaborated further 
by Cremona. [63] The graphical calculus has been applied 
by 0. Mohr of Dresden to the elastic line for continuous 
spans. Henry T. Eddy, of the Rose Polytechnic Institute, 
gives graphical solutions of problems on the maximum stresses 
in bridges under concentrated loads, with aid of what he calls 
"reaction polygons." A standard work, La Statique graphique, 
1874, was issued by Maurice Levy of Paris. 

Descriptive geometry (reduced to a science by Monge in 
France, and elaborated further by his successors, Hachette, 
Dupin, Olivier, J. de la Gournerie) was soon studied also 
in other countries. The French directed their attention 
mainly to the theory of surfaces and their curvature; the 
Germans and Swiss, through Schreiber, Pohlke, Schlessinger, 
and particularly Fiedler, interwove projective and descriptive 
geometry. Bellavitis in Italy worked along the same line. The 
theory of shades and shadows was first investigated by the 
French writers just quoted, and in Germany treated most 


exhaustively by Burmester. [62] 

During the present century very remarkable generalisations 
have been made, which reach to the very root of two of 
the oldest branches of mathematics, — elementary algebra and 
geometry. In algebra the laws of operation have been extended; 
in geometry the axioms have been searched to the bottom, 
and the conclusion has been reached that the space defined 
by Euclid's axioms is not the only possible non-contradictory 
space. Euclid proved (I. 27) that "if a straight line falling 
on two other straight lines make the alternate angles equal 
to one another, the two straight lines shall be parallel to one 
another." Being unable to prove that in every other case 
the two lines are not parallel, he assumed this to be true in 
what is generally called the 12th "axiom," by some the 11th 
"axiom." But this so-called axiom is far from axiomatic. 
After centuries of desperate but fruitless attempts to prove 
Euclid's assumption, the bold idea dawned upon the minds 
of several mathematicians that a geometry might be built 
up without assuming the parallel-axiom. While Legendre 
still endeavoured to establish the axiom by rigid proof, 
Lobatchewsky brought out a publication which assumed the 
contradictory of that axiom, and which was the first of a series 
of articles destined to clear up obscurities in the fundamental 
concepts, and to greatly extend the field of geometry. 

Nicholaus Ivanovitch Lobatchewsky (1793-1856) was 
born at Makarief, in Nischni-Nowgorod, Russia, studied at 
Kasan, and from 1827 to 1846 was professor and rector of the 
University of Kasan. His views on the foundation of geometry 


were first made public in a discourse before the physical 
and mathematical faculty at Kasan, and first printed in the 
Kasan Messenger for 1829, and then in the Gelehrte Schriften 
der Universitdt Kasan, 1836-1838, under the title, "New 
Elements of Geometry, with a complete theory of Parallels." 
Being in the Russian language, the work remained unknown 
to foreigners, but even at home it attracted no notice. In 
1840 he published a brief statement of his researches in Berlin. 
Lobatchewsky constructed an "imaginary geometry," as he 
called it, which has been described by Clifford as "quite 
simple, merely Euclid without the vicious assumption." A 
remarkable part of this geometry is this, that through a point 
an indefinite number of lines can be drawn in a plane, none 
of which cut a given line in the same plane. A similar system 
of geometry was deduced independently by the Bolyais in 
Hungary, who called it "absolute geometry." 

Wolfgang Bolyai de Bolya (1775-1856) was born in 
Szekler-Land, Transylvania. After studying at Jena, he went 
to Gottingen, where he became intimate with Gauss, then 
nineteen years old. Gauss used to say that Bolyai was the 
only man who fully understood his views on the metaphysics 
of mathematics. Bolyai became professor at the Reformed 
College of Maros-Vasarhely, where for forty-seven years he had 
for his pupils most of the present professors of Transylvania. 
The first publications of this remarkable genius were dramas 
and poetry. Clad in old-time planter's garb, he was truly 
original in his private life as well as in his mode of thinking. 
He was extremely modest. No monument, said he, should 


stand over his grave, only an apple-tree, in memory of the 
three apples; the two of Eve and Paris, which made hell out of 
earth, and that of Newton, which elevated the earth again into 
the circle of heavenly bodies. [64] His son, Johann Bolyai 
(1802-1860), was educated for the army, and distinguished 
himself as a profound mathematician, an impassioned violin- 
player, and an expert fencer. He once accepted the challenge 
of thirteen officers on condition that after each duel he might 
play a piece on his violin, and he vanquished them all. 

The chief mathematical work of Wolfgang Bolyai appeared 
in two volumes, 1832-1833, entitled Tentamen juventutem 
studiosam in elementa matheseos puree. . . introducendi. It is 
followed by an appendix composed by his son Johann on The 
Science Absolute of Space. Its twenty-six pages make the 
name of Johann Bolyai immortal. He published nothing else, 
but he left behind one thousand pages of manuscript which 
have never been read by a competent mathematician! His 
father seems to have been the only person in Hungary who 
really appreciated the merits of his son's work. For thirty- 
five years this appendix, as also Lobatchewsky's researches, 
remained in almost entire oblivion. Finally Richard Baltzer 
of the University of Giessen, in 1867, called attention to 
the wonderful researches. Johann Bolyai's Science Absolute 
of Space and Lobatchewsky's Geometrical Researches on the 
Theory of Parallels (1840) were rendered easily accessible to 
American readers by translations into English made in 1891 
by George Bruce Halsted of the University of Texas. 

The Russian and Hungarian mathematicians were not 


the only ones to whom pangeometry suggested itself. A 
copy of the Tentamen reached Gauss, the elder Bolyai's 
former room-mate at Gottingen, and this Nestor of German 
mathematicians was surprised to discover in it worked out 
what he himself had begun long before, only to leave it 
after him in his papers. As early as 1792 he had started 
on researches of that character. His letters show that in 
1799 he was trying to prove a priori the reality of Euclid's 
system; but some time within the next thirty years he arrived 
at the conclusion reached by Lobatchewsky and Bolyai. In 
1829 he wrote to Bessel, stating that his "conviction that 
we cannot found geometry completely a priori has become, 
if possible, still firmer," and that "if number is merely a 
product of our mind, space has also a reality beyond our mind 
of which we cannot fully foreordain the laws a priori." The 
term non-Euclidean geometry is due to Gauss. It has recently 
been brought to notice that Geronimo Saccheri, a Jesuit 
father of Milan, in 1733 anticipated Lobatchewsky's doctrine 
of the parallel angle. Moreover, G. B. Halsted has pointed 
out that in 1766 Lambert wrote a paper "Zur Theorie der 
Parallellinien," published in the Leipziger Magazin fur reine 
und angewandte Mathematik, 1786, in which: (1) The failure 
of the parallel- axiom in surface-spherics gives a geometry with 
angle-sum > 2 right angles; (2) In order to make intuitive a 
geometry with angle-sum < 2 right angles we need the aid 
of an "imaginary sphere" (pseudo-sphere); (3) In a space 
with the angle-sum differing from 2 right angles, there is an 
absolute measure (Bolyai's natural unit for length). 


In 1854, nearly twenty years later, Gauss heard from 
his pupil, Riemann, a marvellous dissertation carrying the 
discussion one step further by developing the notion of n-ply 
extended magnitude, and the measure-relations of which a 
manifoldness of n dimensions is capable, on the assumption 
that every line may be measured by every other. Riemann 
applied his ideas to space. He taught us to distinguish 
between "unboundedness" and "infinite extent." According 
to him we have in our mind a more general notion of 
space, i.e. a notion of non- Euclidean space; but we learn by 
experience that our physical space is, if not exactly, at least 
to high degree of approximation, Euclidean space. Riemann's 
profound dissertation was not published until 1867, when 
it appeared in the Gottingen Abhandlungen. Before this 
the idea of n-dimensions had suggested itself under various 
aspects to Lagrange, Plucker, and H. Grassmann. About 
the same time with Riemann's paper, others were published 
from the pens of Helmholtz and Beltrami. These contributed 
powerfully to the victory of logic over excessive empiricism. 
This period marks the beginning of lively discussions upon 
this subject. Some writers — Bellavitis, for example — were 
able to see in non-Euclidean geometry and n-dimensional 
space nothing but huge caricatures, or diseased outgrowths 
of mathematics. Helmholtz's article was entitled Thatsachen, 
welche der Geometrie zu Grunde liegen, 1868, and contained 
many of the ideas of Riemann. Helmholtz popularised the 
subject in lectures, and in articles for various magazines. 

Eugenio Beltrami, born at Cremona, Italy, in 1835, and 


now professor at Rome, wrote the classical paper Saggio 
di interpretazione della geometria non-euclidea (Giorn. di 
Matem., 6), which is analytical (and, like several other papers, 
should be mentioned elsewhere were we to adhere to a strict 
separation between synthesis and analysis). He reached the 
brilliant and surprising conclusion that the theorems of non- 
Euclidean geometry find their realisation upon surfaces of 
constant negative curvature. He studied, also, surfaces of 
constant positive curvature, and ended with the interesting 
theorem that the space of constant positive curvature is 
contained in the space of constant negative curvature. These 
researches of Beltrami, Helmholtz, and Riemann culminated 
in the conclusion that on surfaces of constant curvature we 
may have three geometries, — the non-Euclidean on a surface 
of constant negative curvature, the spherical on a surface 
of constant positive curvature, and the Euclidean geometry 
on a surface of zero curvature. The three geometries do 
not contradict each other, but are members of a system, — a 
geometrical trinity. The ideas of hyperspace were brilliantly 
expounded and popularised in England by Clifford. 

William Kingdon Clifford (1845-1879) was born at Ex- 
eter, educated at Trinity College, Cambridge, and from 1871 
until his death professor of applied mathematics in University 
College, London. His premature death left incomplete several 
brilliant researches which he had entered upon. Among 
these are his paper On Classification of Loci and his Theory 
of Graphs. He wrote articles On the Canonical Form and 
Dissection of a Riemann's Surface, on Biquaternions, and an 


incomplete work on the Elements of Dynamic. The theory of 
polars of curves and surfaces was generalised by him and by 
Reye. His classification of loci, 1878, being a general study 
of curves, was an introduction to the study of n-dimensional 
space in a direction mainly projective. This study has been 
continued since chiefly by G. Veronese of Padua, C. Segre of 
Turin, E. Bertini, F. Aschieri, P. Del Pezzo of Naples. 

Beltrami's researches on non-Euclidean geometry were 
followed, in 1871, by important investigations of Felix Klein, 
resting upon Cayley's Sixth Memoir on Quantics, 1859. The 
question whether it is not possible to so express the metrical 
properties of figures that they will not vary by projection (or 
linear transformation) had been solved for special projections 
by Chasles, Poncelet, and E. Laguerre (1834-1886) of Paris, 
but it remained for Cayley to give a general solution by defining 
the distance between two points as an arbitrary constant 
multiplied by the logarithm of the anharmonic ratio in which 
the line joining the two points is divided by the fundamental 
quadric. Enlarging upon this notion, Klein showed the 
independence of projective geometry from the parallel- axiom, 
and by properly choosing the law of the measurement of 
distance deduced from projective geometry the spherical, 
Euclidean, and pseudospherical geometries, named by him 
respectively the elliptic, parabolic, and hyperbolic geometries. 
This suggestive investigation was followed up by numerous 
writers, particularly by G. Battaglini of Naples, E. d'Ovidio 
of Turin, R. de Paolis of Pisa, F. Aschieri, A. Cayley, 
F. Lindemann of Munich, E. Scheringof Gottingen, W. Story of 


Clark University, H. Stahl of Tubingen, A. Voss of Wiirzburg, 
Homersham Cox, A. Buchheim. [55] The geometry of n 
dimensions was studied along a line mainly metrical by a host 
of writers, among whom may be mentioned Simon Newcomb 
of the Johns Hopkins University, L. Schlafli of Bern, W. I. 
Stringham of the University of California, W. Killing of 
Minister, T. Craig of the Johns Hopkins, R. Lipschitz of Bonn. 
R. S. Heath and Killing investigated the kinematics and 
mechanics of such a space. Regular solids in n-dimensional 
space were studied by Stringham, Ellery W. Davis of the 
University of Nebraska, R. Hoppe of Berlin, and others. 
Stringham gave pictures of projections upon our space of 
regular solids in four dimensions, and Schlegel at Hagen 
constructed models of such projections. These are among the 
most curious of a series of models published by L. Brill in 
Darmstadt. It has been pointed out that if a fourth dimension 
existed, certain motions could take place which we hold to be 
impossible. Thus Newcomb showed the possibility of turning 
a closed material shell inside out by simple flexure without 
either stretching or tearing; Klein pointed out that knots 
could not be tied; Veronese showed that a body could be 
removed from a closed room without breaking the walls; C. S. 
Peirce proved that a body in four-fold space either rotates 
about two axes at once, or cannot rotate without losing one of 
its dimensions. 



In the preceding chapter we endeavoured to give a flash- 
light view of the rapid advance of synthetic geometry. In 
connection with hyperspace we also mentioned analytical 
treatises. Modern synthetic and modern analytical geometry 
have much in common, and may be grouped together under the 
common name "projective geometry." Each has advantages 
over the other. The continual direct viewing of figures as 
existing in space adds exceptional charm to the study of the 
former, but the latter has the advantage in this, that a well- 
established routine in a certain degree may outrun thought 
itself, and thereby aid original research. While in Germany 
Steiner and von Staudt developed synthetic geometry, Plucker 
laid the foundation of modern analytic geometry. 

Julius Plucker (1801-1868) was born at Elberfeld, in 
Prussia. After studying at Bonn, Berlin, and Heidelberg, he 
spent a short time in Paris attending lectures of Monge and his 
pupils. Between 1826 and 1836 he held positions successively 
at Bonn, Berlin, and Halle. He then became professor 
of physics at Bonn. Until 1846 his original researches 
were on geometry. In 1828 and in 1831 he published 
his Analytisch-Geometrische Entwicklungen in two volumes. 
Therein he adopted the abbreviated notation (used before 
him in a more restricted way by Bobillier), and avoided 
the tedious process of algebraic elimination by a geometric 
consideration. In the second volume the principle of duality is 
formulated analytically. With him duality and homogeneity 


found expression already in his system of co-ordinates. The 
homogenous or tri- linear system used by him is much the same 
as the co-ordinates of Mobius. In the identity of analytical 
operation and geometric construction Plucker looked for 
the source of his proofs. The System der Analytischen 
Geometrie, 1835, contains a complete classification of plane 
curves of the third order, based on the nature of the points 
at infinity. The Theorie der Algebraischen Curven, 1839, 
contains, besides an enumeration of curves of the fourth order, 
the analytic relations between the ordinary singularities of 
plane curves known as "Plucker's equations," by which he was 
able to explain "Poncelet's paradox." The discovery of these 
relations is, says Cayley, "the most important one beyond all 
comparison in the entire subject of modern geometry." But 
in Germany Plucker's researches met with no favour. His 
method was declared to be unproductive as compared with the 
synthetic method of Steiner and Poncelet! His relations with 
Jacobi were not altogether friendly. Steiner once declared that 
he would stop writing for Crelle 's Journal if Plucker continued 
to contribute to it. [66] The result was that many of Plucker's 
researches were published in foreign journals, and that his 
work came to be better known in France and England than 
in his native country. The charge was also brought against 
Plucker that, though occupying the chair of physics, he was 
no physicist. This induced him to relinquish mathematics, 
and for nearly twenty years to devote his energies to physics. 
Important discoveries on Fresnel's wave-surface, magnetism, 
spectrum-analysis were made by him. But towards the close 


of his life he returned to his first love, — mathematics, — and 
enriched it with new discoveries. By considering space as made 
up of lines he created a "new geometry of space." Regarding a 
right line as a curve involving four arbitrary parameters, one 
has the whole system of lines in space. By connecting them by 
a single relation, he got a "complex" of lines; by connecting 
them with a twofold relation, he got a "congruency" of lines. 
His first researches on this subject were laid before the Royal 
Society in 1865. His further investigations thereon appeared 
in 1868 in a posthumous work entitled Neue Geometrie des 
Raumes gegrilndet auf die Betrachtung der geraden Linie als 
Raumelement, edited by Felix Klein. Plucker's analysis lacks 
the elegance found in Lagrange, Jacobi, Hesse, and Clebsch. 
For many years he had not kept up with the progress of 
geometry, so that many investigations in his last work had 
already received more general treatment on the part of others. 
The work contained, nevertheless, much that was fresh and 
original. The theory of complexes of the second degree, left 
unfinished by Plucker, was continued by Felix Klein, who 
greatly extended and supplemented the ideas of his master. 

Ludwig Otto Hesse (1811-1874) was born at Konigsberg, 
and studied at the university of his native place under Bessel, 
Jacobi, Richelot, and F. Neumann. Having taken the 
doctor's degree in 1840, he became docent at Konigsberg, 
and in 1845 extraordinary professor there. Among his 
pupils at that time were Durege, Carl Neumann, Clebsch, 
Kirchhoff. The Konigsberg period was one of great activity 
for Hesse. Every new discovery increased his zeal for still 


greater achievement. His earliest researches were on surfaces 
of the second order, and were partly synthetic. He solved the 
problem to construct any tenth point of such a surface when 
nine points are given. The analogous problem for a conic had 
been solved by Pascal by means of the hexagram. A difficult 
problem confronting mathematicians of this time was that of 
elimination. Phicker had seen that the main advantage of 
his special method in analytic geometry lay in the avoidance 
of algebraic elimination. Hesse, however, showed how by 
determinants to make algebraic elimination easy. In his 
earlier results he was anticipated by Sylvester, who published 
his dialytic method of elimination in 1840. These advances in 
algebra Hesse applied to the analytic study of curves of the 
third order. By linear substitutions, he reduced a form of the 
third degree in three variables to one of only four terms, and 
was led to an important determinant involving the second 
differential coefficient of a form of the third degree, called the 
"Hessian." The "Hessian" plays a leading part in the theory 
of invariants, a subject first studied by Cayley. Hesse showed 
that his determinant gives for every curve another curve, such 
that the double points of the first are points on the second, or 
"Hessian." Similarly for surfaces (Crelle, 1844). Many of the 
most important theorems on curves of the third order are due 
to Hesse. He determined the curve of the 14th order, which 
passes through the 56 points of contact of the 28 bi-tangents of 
a curve of the fourth order. His great memoir on this subject 
(Crelle, 1855) was published at the same time as was a paper 
by Steiner treating of the same subject. 


Hesse's income at Konigsberg had not kept pace with his 
growing reputation. Hardly was he able to support himself 
and family. In 1855 he accepted a more lucrative position 
at Halle, and in 1856 one at Heidelberg. Here he remained 
until 1868, when he accepted a position at a technic school in 
Munich. [67] At Heidelberg he revised and enlarged upon his 
previous researches, and published in 1861 his Vorlesungen 
iiber die Analytische Geometrie des Raumes, insbesondere 
iiber Fldchen 2. Ordnung. More elementary works soon 
followed. While in Heidelberg he elaborated a principle, his 
"Uebertragungsprincip." According to this, there corresponds 
to every point in a plane a pair of points in a line, and the 
projective geometry of the plane can be carried back to the 
geometry of points in a line. 

The researches of Pliicker and Hesse were continued in 
England by Cayley, Salmon, and Sylvester. It may be premised 
here that among the early writers on analytical geometry in 
England was James Booth (1806-1878), whose chief results 
are embodied in his Treatise on Some New Geometrical 
Methods] and James MacCullagh (1809-1846), who was 
professor of natural philosophy at Dublin, and made some 
valuable discoveries on the theory of quadrics. The influence 
of these men on the progress of geometry was insignificant, for 
the interchange of scientific results between different nations 
was not so complete at that time as might have been desired. In 
further illustration of this, we mention that Chasles in France 
elaborated subjects which had previously been disposed of by 
Steiner in Germany, and Steiner published researches which 


had been given by Cayley, Sylvester, and Salmon nearly 
five years earlier. Cayley and Salmon in 1849 determined 
the straight lines in a cubic surface, and studied its principal 
properties, while Sylvester in 1851 discovered the pentahedron 
of such a surface. Cayley extended Plucker's equations to 
curves of higher singularities. Cayley's own investigations, 
and those of M. Nother of Erlangen, G. H. Halphen (1844- 
1889) of the Polytechnic School in Paris, De La Gournerie 
of Paris, A. Brill of Tubingen, lead to the conclusion that 
each higher singularity of a curve is equivalent to a certain 
number of simple singularities, — the node, the ordinary cusp, 
the double tangent, and the inflection. Sylvester studied the 
"twisted Cartesian," a curve of the fourth order. Salmon 
helped powerfully towards the spreading of a knowledge of 
the new algebraic and geometric methods by the publication 
of an excellent series of text-books ( Conic Sections, Modern 
Higher Algebra, Higher Plane Curves, Geometry of Three 
Dimensions), which have been placed within easy reach of 
German readers by a free translation, with additions, made 
by Wilhelm Fiedler of the Polytechnicum in Zurich. The next 
great worker in the field of analytic geometry was Clebsch. 

Rudolf Friedrich Alfred Clebsch (1833-1872) was born 
at Konigsberg in Prussia, studied at the university of that place 
under Hesse, Richelot, F. Neumann. From 1858 to 1863 he 
held the chair of theoretical mechanics at the Polytechnicum 
in Carlsruhe. The study of Salmon's works led him into 
algebra and geometry. In 1863 he accepted a position at the 
University of Giessen, where he worked in conjunction with 


Paul Gordan (now of Erlangen). In 1868 Clebsch went to 
Gottingen, and remained there until his death. He worked 
successively at the following subjects: Mathematical physics, 
the calculus of variations and partial differential equations 
of the first order, the general theory of curves and surfaces, 
Abelian functions and their use in geometry, the theory of 
invariants, and "Flachenabbildung." [68] He proved theorems 
on the pentahedron enunciated by Sylvester and Steiner; 
he made systematic use of "deficiency" (Geschlecht) as a 
fundamental principle in the classification of algebraic curves. 
The notion of deficiency was known before him to Abel and 
Riemann. At the beginning of his career, Clebsch had shown 
how elliptic functions could be advantageously applied to 
Malfatti's problem. The idea involved therein, viz. the use 
of higher transcendentals in the study of geometry, led him 
to his greatest discoveries. Not only did he apply Abelian 
functions to geometry, but conversely, he drew geometry into 
the service of Abelian functions. 

Clebsch made liberal use of determinants. His study of 
curves and surfaces began with the determination of the points 
of contact of lines which meet a surface in four consecutive 
points. Salmon had proved that these points lie on the 
intersection of the surface with a derived surface of the degree 
lln — 24, but his solution was given in inconvenient form. 
Clebsch's investigation thereon is a most beautiful piece of 

The representation of one surface upon another (Flachen- 
abbildung), so that they have a (1,1) correspondence, was 


thoroughly studied for the first time by Clebsch. The rep- 
resentation of a sphere on a plane is an old problem which 
drew the attention of Ptolemaeus, Gerard Mercator, Lambert, 
Gauss, Lagrange. Its importance in the construction of maps 
is obvious. Gauss was the first to represent a surface upon 
another with a view of more easily arriving at its properties. 
Pliicker, Chasles, Cayley, thus represented on a plane the 
geometry of quadric surfaces; Clebsch and Cremona, that of 
cubic surfaces. Other surfaces have been studied in the same 
way by recent writers, particularly M. Nother of Erlangen, 
Armenante, Felix Klein, Korndorfer, Caporali, H. G. Zeuthen 
of Copenhagen. A fundamental question which has as yet 
received only a partial answer is this: What surfaces can be 
represented by a (1, 1) correspondence upon a given surface? 
This and the analogous question for curves was studied by 
Clebsch. Higher correspondences between surfaces have been 
investigated by Cayley and Nother. The theory of surfaces 
has been studied also by Joseph Alfred Serret (1819-1885), 
professor at the Sorbonne in Paris, Jean Gaston Darboux 
of Paris, John Casey of Dublin (died 1891), W. R. W. Roberts 
of Dublin, H. Schroter (1829-1892) of Breslau. Surfaces of 
the fourth order were investigated by Kummer, and Fresnel's 
wave-surface, studied by Hamilton, is a particular case of 
Rummer's quartic surface, with sixteen canonical points and 
sixteen singular tangent planes. [56] 

The infinitesimal calculus was first applied to the determi- 
nation of the measure of curvature of surfaces by Lagrange, 
Euler, and Meusnier (1754-1793) of Paris. Then followed the 


researches of Monge and Dupin, but they were eclipsed by 
the work of Gauss, who disposed of this difficult subject in a 
way that opened new vistas to geometricians. His treatment 
is embodied in the Disquisitiones generales circa superfi- 
cies curvas (1827) and Untersuchungen iiber gegenstande der 
hoheren Geoddsie of 1843 and 1846. He defined the measure 
of curvature at a point to be the reciprocal of the product 
of the two principal radii of curvature at that point. From 
this flows the theorem of Johann August Grunert (1797-1872; 
professor in Greifswald), that the arithmetical mean of the 
radii of curvature of all normal sections through a point is 
the radius of a sphere which has the same measure of cur- 
vature as has the surface at that point. Gauss's deduction 
of the formula of curvature was simplified through the use 
of determinants by Heinrich Richard Baltzer (1818-1887) of 
Giessen. [69] Gauss obtained an interesting theorem that if 
one surface be developed (abgewickelt) upon another, the 
measure of curvature remains unaltered at each point. The 
question whether two surfaces having the same curvature in 
corresponding points can be unwound, one upon the other, 
was answered by F. Minding in the affirmative only when the 
curvature is constant. The case of variable curvature is diffi- 
cult, and was studied by Minding, J. Liouville (1806-1882) of 
the Polytechnic School in Paris, Ossian Bonnet of Paris (died 
1892). Gauss's measure of curvature, expressed as a function 
of curvilinear co-ordinates, gave an impetus to the study of 
differential-invariants, or differential-parameters, which have 
been investigated by Jacobi, C. Neumann, Sir James Cockle, 


Halphen, and elaborated into a general theory by Beltrami, 
S. Lie, and others. Beltrami showed also the connection 
between the measure of curvature and the geometric axioms. 

Various researches have been brought under the head 
of "analysis situs." The subject was first investigated by 
Leibniz, and was later treated by Gauss, whose theory of 
knots ( Verschlingungen) has been employed recently by J. B. 
Listing, O. Simony, F. Dingeldey, and others in their "topologic 
studies." Tait was led to the study of knots by Sir William 
Thomson's theory of vortex atoms. In the hands of Riemann 
the analysis situs had for its object the determination of what 
remains unchanged under transformations brought about by 
a combination of infinitesimal distortions. In continuation of 
his work, Walter Dyck of Munich wrote on the analysis situs 
of three-dimensional spaces. 

Of geometrical text-books not yet mentioned, reference 
should be made to Alfred Clebsch's Vorlesungen ilber 
Geometrie, edited by Ferdinand Lindemann, now of Munich; 
Frost's Solid Geometry; Durege's Ebene Curven dritter 


The progress of algebra in recent times may be considered 
under three principal heads: the study of fundamental laws 
and the birth of new algebras, the growth of the theory of 
equations, and the development of what is called modern 
higher algebra. 


We have already spoken of George Peacock and D. F. 
Gregory in connection with the fundamental laws of algebra. 
Much was done in this line by De Morgan. 

Augustus De Morgan (1806-1871) was born at Madura 
(Madras), and educated at Trinity College, Cambridge. 
His scruples about the doctrines of the established church 
prevented him from proceeding to the M.A. degree, and from 
sitting for a fellowship. In 1828 he became professor at the 
newly established University of London, and taught there 
until 1867, except for five years, from 1831-1835. De Morgan 
was a unique, manly character, and pre-eminent as a teacher. 
The value of his original work lies not so much in increasing 
our stock of mathematical knowledge as in putting it all upon 
a thoroughly logical basis. He felt keenly the lack of close 
reasoning in mathematics as he received it. He said once: 
"We know that mathematicians care no more for logic than 
logicians for mathematics. The two eyes of exact science are 
mathematics and logic: the mathematical sect puts out the 
logical eye, the logical sect puts out the mathematical eye; 
each believing that it can see better with one eye than with 
two." De Morgan saw with both eyes. He analysed logic 
mathematically, and studied the logical analysis of the laws, 
symbols, and operations of mathematics; he wrote a Formal 
Logic as well as a Double Algebra, and corresponded both with 
Sir William Hamilton, the metaphysician, and Sir William 
Rowan Hamilton, the mathematician. Few contemporaries 
were as profoundly read in the history of mathematics as 
was De Morgan. No subject was too insignificant to receive 


his attention. The authorship of "Cocker's Arithmetic" and 
the work of circle-squarers was investigated as minutely as 
was the history of the invention of the calculus. Numerous 
articles of his lie scattered in the volumes of the Penny and 
English Cyclopedias. His Differential Calculus, 1842, is still 
a standard work, and contains much that is original with the 
author. For the Encyclopedia Metropolitana he wrote on the 
calculus of functions (giving principles of symbolic reasoning) 
and on the theory of probability. Celebrated is his Budget of 
Paradoxes, 1872. He published memoirs "On the Foundation 
of Algebra" (Trans, of Cam. Phil. Soc, 1841, 1842, 1844, 
and 1847). 

In Germany symbolical algebra was studied by Martin 
Ohm, who wrote a System der Mathematik in 1822. The 
ideas of Peacock and De Morgan recognise the possibility of 
algebras which differ from ordinary algebra. Such algebras 
were indeed not slow in forthcoming, but, like non- Euclidean 
geometry, some of them were slow in finding recognition. This 
is true of Grassmann's, Bellavitis's, and Peirce's discoveries, 
but Hamilton's quaternions met with immediate appreciation 
in England. These algebras offer a geometrical interpretation 
of imaginaries. During the times of Descartes, Newton, and 
Euler, we have seen the negative and the imaginary, a/— 1, 
accepted as numbers, but the latter was still regarded as an 
algebraic fiction. The first to give it a geometric picture, 
analogous to the geometric interpretation of the negative, 
was H. Kiihn, a teacher in Danzig, in a publication of 1750- 
1751. He represented ay/—l by a line perpendicular to the 


line a, and equal to a in length, and construed \/—l as the 
mean proportional between +1 and — 1. This same idea was 
developed further, so as to give a geometric interpretation 
of a + \/—b, by Jean-Robert Argand (1768-?) of Geneva, in 
a remarkable Essai (1806). [70] The writings of Kiihn and 
Argand were little noticed, and it remained for Gauss to break 
down the last opposition to the imaginary. He introduced i as 
an independent unit co-ordinate to 1, and a + ib as a "complex 
number." The connection between complex numbers and 
points on a plane, though artificial, constituted a powerful aid 
in the further study of symbolic algebra. The mind required a 
visual representation to aid it. The notion of what we now call 
vectors was growing upon mathematicians, and the geometric 
addition of vectors in space was discovered independently by 
Hamilton, Grassmann, and others, about the same time. 

William Rowan Hamilton (1805-1865) was born of 
Scotch parents in Dublin. His early education, carried on at 
home, was mainly in languages. At the age of thirteen he 
is said to have been familiar with as many languages as he 
had lived years. About this time he came across a copy of 
Newton's Universal Arithmetic. After reading that, he took 
up successively analytical geometry, the calculus, Newton's 
Principia, Laplace's Mecanique Celeste. At the age of eighteen 
he published a paper correcting a mistake in Laplace's work. 
In 1824 he entered Trinity College, Dublin, and in 1827, while 
he was still an undergraduate, he was appointed to the chair 
of astronomy. His early papers were on optics. In 1832 he 
predicted conical refraction, a discovery by aid of mathematics 


which ranks with the discovery of Neptune by Le Verrier and 
Adams. Then followed papers on the Principle of Varying 
Action (1827) and a general method of dynamics (1834-1835). 
He wrote also on the solution of equations of the fifth degree, 
the hodograph, fluctuating functions, the numerical solution 
of differential equations. 

The capital discovery of Hamilton is his quaternions, in 
which his study of algebra culminated. In 1835 he published 
in the Transactions of the Royal Irish Academy his Theory of 
Algebraic Couples. He regarded algebra "as being no mere art, 
nor language, nor primarily a science of quantity, but rather 
as the science of order of progression." Time appeared to 
him as the picture of such a progression. Hence his definition 
of algebra as "the science of pure time." It was the subject 
of years' meditation for him to determine what he should 
regard as the product of each pair of a system of perpendicular 
directed lines. At last, on the 16th of October, 1843, while 
walking with his wife one evening, along the Royal Canal 
in Dublin, the discovery of quaternions flashed upon him, 
and he then engraved with his knife on a stone in Brougham 
Bridge the fundamental formula i 2 = j 2 = k 2 = ijk = — 1. 
At the general meeting of the Irish Academy, a month 
later, he made the first communication on quaternions. An 
account of the discovery was given the following year in 
the Philosophical Magazine. Hamilton displayed wonderful 
fertility in their development. His Lectures on Quaternions, 
delivered in Dublin, were printed in 1852. His Elements of 
Quaternions appeared in 1866. Quaternions were greatly 


admired in England from the start, but on the Continent 
they received less attention. P. G. Tait's Elementary Treatise 
helped powerfully to spread a knowledge of them in England. 
Cayley, Clifford, and Tait advanced the subject somewhat by 
original contributions. But there has been little progress in 
recent years, except that made by Sylvester in the solution of 
quaternion equations, nor has the application of quaternions 
to physics been as extended as was predicted. The change in 
notation made in France by Hoiiel and by Laisant has been 
considered in England as a wrong step, but the true cause 
for the lack of progress is perhaps more deep-seated. There 
is indeed great doubt as to whether the quaternionic product 
can claim a necessary and fundamental place in a system 
of vector analysis. Physicists claim that there is a loss of 
naturalness in taking the square of a vector to be negative. 
In order to meet more adequately their wants, J. W. Gibbs 
of Yale University and A. Macfarlane of the University of 
Texas, have each suggested an algebra of vectors with a new 
notation. Each gives a definition of his own for the product of 
two vectors, but in such a way that the square of a vector is 
positive. A third system of vector analysis has been used by 
Oliver Heaviside in his electrical researches. 

Hermann Grassmann (1809-1877) was born at Stettin, 
attended a gymnasium at his native place (where his father was 
teacher of mathematics and physics), and studied theology 
in Berlin for three years. In 1834 he succeeded Steiner as 
teacher of mathematics in an industrial school in Berlin, but 
returned to Stettin in 1836 to assume the duties of teacher 


of mathematics, the sciences, and of religion in a school 
there. [71] Up to this time his knowledge of mathematics 
was pretty much confined to what he had learned from his 
father, who had written two books on "Raumlehre" and 
"Grossenlehre." But now he made his acquaintance with the 
works of Lacroix, Lagrange, and Laplace. He noticed that 
Laplace's results could be reached in a shorter way by some 
new ideas advanced in his father's books, and he proceeded to 
elaborate this abridged method, and to apply it in the study 
of tides. He was thus led to a new geometric analysis. In 
1840 he had made considerable progress in its development, 
but a new book of Schleiermacher drew him again to theology. 
In 1842 he resumed mathematical research, and becoming 
thoroughly convinced of the importance of his new analysis, 
decided to devote himself to it. It now became his ambition 
to secure a mathematical chair at a university, but in this he 
never succeeded. In 1844 appeared his great classical work, 
the Lineale Ausdehnungslehre, which was full of new and 
strange matter, and so general, abstract, and out of fashion 
in its mode of exposition, that it could hardly have had less 
influence on European mathematics during its first twenty 
years, had it been published in China. Gauss, Grunert, and 
Mobius glanced over it, praised it, but complained of the 
strange terminology and its "philosophische Allgemeinheit." 
Eight years afterwards, Bretschneider of Gotha was said to be 
the only man who had read it through. An article in Crelle 's 
Journal, in which Grassmann eclipsed the geometers of that 
time by constructing, with aid of his method, geometrically 


any algebraic curve, remained again unnoticed. Need we 
marvel if Grassmann turned his attention to other subjects,— 
to Schleiermacher's philosophy, to politics, to philology? Still, 
articles by him continued to appear in Crelle 's Journal, and in 
1862 came out the second part of his Ausdehnungslehre. It was 
intended to show better than the first part the broad scope 
of the Ausdehnungslehre, by considering not only geometric 
applications, but by treating also of algebraic functions, 
infinite series, and the differential and integral calculus. But 
the second part was no more appreciated than the first. 
At the age of fifty-three, this wonderful man, with heavy 
heart, gave up mathematics, and directed his energies to the 
study of Sanskrit, achieving in philology results which were 
better appreciated, and which vie in splendour with those in 

Common to the Ausdehnungslehre and to quaternions are 
geometric addition, the function of two vectors represented 
in quaternions by Sa(3 and Va(3, and the linear vector func- 
tions. The quaternion is peculiar to Hamilton, while with 
Grassmann we find in addition to the algebra of vectors a geo- 
metrical algebra of wide application, and resembling Mobius's 
Barycentrische Calcul, in which the point is the fundamental 
element. Grassmann developed the idea of the "external 
product," the "internal product," and the "open product." 
The last we now call a matrix. His Ausdehnungslehre has 
very great extension, having no limitation to any particular 
number of dimensions. Only in recent years has the wonderful 
richness of his discoveries begun to be appreciated. A second 


edition of the Ausdehnungslehre of 1844 was printed in 1877. 
C. S. Peirce gave a representation of Grassmann's system in 
the logical notation, and E. W. Hyde of the University of 
Cincinnati wrote the first text-book on Grassmann's calculus 
in the English language. 

Discoveries of less value, which in part covered those 
of Grassmann and Hamilton, were made by Saint- Venant 
(1797-1886), who described the multiplication of vectors, 
and the addition of vectors and oriented areas; by Cauchy, 
whose "clefs algebriques" were units subject to combinatorial 
multiplication, and were applied by the author to the theory 
of elimination in the same way as had been done earlier 
by Grassmann; by Justus Bellavitis (1803-1880), who 
published in 1835 and 1837 in the Annali delle Scienze his 
calculus of aequipollences. Bellavitis, for many years professor 
at Padua, was a self-taught mathematician of much power, 
who in his thirty-eighth year laid down a city office in his native 
place, Bassano, that he might give his time to science. [65] 

The first impression of Grassmann's ideas is marked in the 
writings of Hermann Hankel (1839-1873), who published 
in 1867 his Vorlesungen iiber die Complexen Zahlen. Hankel, 
then docent in Leipzig, had been in correspondence with 
Grassmann. The "alternate numbers" of Hankel are subject 
to his law of combinatorial multiplication. In considering 
the foundations of algebra Hankel affirms the principle of the 
permanence of formal laws previously enunciated incompletely 
by Peacock. Hankel was a close student of mathematical 
history, and left behind an unfinished work thereon. Before 


his death he was professor at Tubingen. His Complexen Zahlen 
was at first little read, and we must turn to Victor Schlegel 
of Hagen as the successful interpreter of Grassmann. Schlegel 
was at one time a young colleague of Grassmann at the 
Marienstifts- Gymnasium in Stettin. Encouraged by Clebsch, 
Schlegel wrote a System der Raumlehre which explained the 
essential conceptions and operations of the Ausdehnungslehre. 
Multiple algebra was powerfully advanced by Peirce, whose 
theory is not geometrical, as are those of Hamilton and 
Grassmann. Benjamin Peirce (1809-1880) was born at 
Salem, Mass., and graduated at Harvard College, having as 
undergraduate carried the study of mathematics far beyond 
the limits of the college course. [2] When Bowditch was 
preparing his translation and commentary of the Mecanique 
Celeste, young Peirce helped in reading the proof-sheets. He 
was made professor at Harvard in 1833, a position which he 
retained until his death. For some years he was in charge 
of the Nautical Almanac and superintendent of the United 
States Coast Survey. He published a series of college text- 
books on mathematics, an Analytical Mechanics, 1855, and 
calculated, together with Sears C. Walker of Washington, 
the orbit of Neptune. Profound are his researches on Linear 
Associative Algebra. The first of several papers thereon was 
read at the first meeting of the American Association for 
the Advancement of Science in 1864. Lithographed copies 
of a memoir were distributed among friends in 1870, but so 
small seemed to be the interest taken in this subject that 
the memoir was not printed until 1881 {Am. Jour. Math., 


Vol. IV., No. 2). Peirce works out the multiplication tables, 
first of single algebras, then of double algebras, and so on up 
to sextuple, making in all 162 algebras, which he shows to be 
possible on the consideration of symbols A, B, etc., which are 
linear functions of a determinate number of letters or units 
i, j, k, I, etc., with coefficients which are ordinary analytical 
magnitudes, real or imaginary, — the letters i, j, etc., being 
such that every binary combination i 2 , ij, ji, etc., is equal 
to a linear function of the letters, but under the restriction 
of satisfying the associative law. [56] Charles S. Peirce, a 
son of Benjamin Peirce, and one of the foremost writers 
on mathematical logic, showed that these algebras were all 
defective forms of quadrate algebras which he had previously 
discovered by logical analysis, and for which he had devised a 
simple notation. Of these quadrate algebras quaternions is a 
simple example; nonions is another. C. S. Peirce showed that 
of all linear associative algebras there are only three in which 
division is unambiguous. These are ordinary single algebra, 
ordinary double algebra, and quaternions, from which the 
imaginary scalar is excluded. He showed that his father's 
algebras are operational and matricular. Lectures on multiple 
algebra were delivered by J. J. Sylvester at the Johns Hopkins 
University, and published in various journals. They treat 
largely of the algebra of matrices. The theory of matrices 
was developed as early as 1858 by Cayley in an important 
memoir which, in the opinion of Sylvester, ushered in the 
reign of Algebra the Second. Clifford, Sylvester, H. Taber, 
C. H. Chapman, carried the investigations much further. 


The originator of matrices is really Hamilton, but his theory, 
published in his Lectures on Quaternions, is less general than 
that of Cayley. The latter makes no reference to Hamilton. 

The theory of determinants [73] was studied by Hoene 
Wronski in Italy and J. Binet in France; but they were fore- 
stalled by the great master of this subject, Cauchy. In a paper 
(Jour, de I'ecole Polyt., IX., 16) Cauchy developed several 
general theorems. He introduced the name determinant, a 
term previously used by Gauss in the functions considered 
by him. In 1826 Jacobi began using this calculus, and he 
gave brilliant proof of its power. In 1841 he wrote extended 
memoirs on determinants in Crelle 's Journal, which rendered 
the theory easily accessible. In England the study of linear 
transformations of quantics gave a powerful impulse. Cayley 
developed skew-determinants and Pfaffians, and introduced 
the use of determinant brackets, or the familiar pair of upright 
lines. More recent researches on determinants appertain to 
special forms. "Continuants" are due to Sylvester; "alter- 
nants," originated by Cauchy, have been developed by Jacobi, 
N. Trudi, H. Nagelbach, and G. Garbieri; "axisymmetric 
determinants," first used by Jacobi, have been studied by 
V. A. Lebesgue, Sylvester, and Hesse; "circulants" are due to 
E. Catalan of Liege, W. Spottiswoode (1825-1883), J. W. L. 
Glaisher, and R. F. Scott; for "centro-symmetric determi- 
nants" we are indebted to G. Zehfuss. E. B. Christoffel 
of Strassburg and G. Frobenius discovered the properties of 
"Wronskians," first used by Wronski. V. Nachreiner and 
S. Giinther, both of Munich, pointed out relations between 


determinants and continued fractions; Scott uses Hankel's al- 
ternate numbers in his treatise. Text-books on determinants 
were written by Spottiswoode (1851), Brioschi (1854), Baltzer 
(1857), Gunther (1875), Dostor (1877), Scott (1880), Muir 
(1882), Hanus (1886). 

Modern higher algebra is especially occupied with the 
theory of linear transformations. Its development is mainly 
the work of Cayley and Sylvester. 

Arthur Cayley, born at Richmond, in Surrey, in 1821, 
was educated at Trinity College, Cambridge. [74] He came 
out Senior Wrangler in 1842. He then devoted some years 
to the study and practice of law. On the foundation of the 
Sadlerian professorship at Cambridge, he accepted the offer 
of that chair, thus giving up a profession promising wealth for 
a very modest provision, but which would enable him to give 
all his time to mathematics. Cayley began his mathematical 
publications in the Cambridge Mathematical Journal while 
he was still an undergraduate. Some of his most brilliant 
discoveries were made during the time of his legal practice. 
There is hardly any subject in pure mathematics which the 
genius of Cayley has not enriched, but most important is his 
creation of a new branch of analysis by his theory of invariants. 
Germs of the principle of invariants are found in the writings 
of Lagrange, Gauss, and particularly of Boole, who showed, in 
1841, that invariance is a property of discriminants generally, 
and who applied it to the theory of orthogonal substitution. 
Cayley set himself the problem to determine a priori what 
functions of the coefficients of a given equation possess this 


property of invariance, and found, to begin with, in 1845, that 
the so-called "hyper-determinants" possessed it. Boole made 
a number of additional discoveries. Then Sylvester began his 
papers in the Cambridge and Dublin Mathematical Journal 
on the Calculus of Forms. After this, discoveries followed in 
rapid succession. At that time Cayley and Sylvester were 
both residents of London, and they stimulated each other by 
frequent oral communications. It has often been difficult to 
determine how much really belongs to each. 

James Joseph Sylvester was born in London in 1814, 
and educated at St. Johns College, Cambridge. He came out 
Second Wrangler in 1837. His Jewish origin incapacitated 
him from taking a degree. In 1846 he became a student at 
the Inner Temple, and was called to the bar in 1850. He 
became professor of natural philosophy at University College, 
London; then, successively, professor of mathematics at the 
University of Virginia, at the Royal Military Academy in 
Woolwich, at the Johns Hopkins University in Baltimore, and 
is, since 1883, professor of geometry at Oxford. His first 
printed paper was on Fresnel's optic theory, 1837. Then 
followed his researches on invariants, the theory of equations, 
theory of partitions, multiple algebra, the theory of numbers, 
and other subjects mentioned elsewhere. About 1874 he 
took part in the development of the geometrical theory of 
link-work movements, originated by the beautiful discovery 
of A. Peaucellier, Capitaine du Genie a Nice (published in 
Nouvelles Annales, 1864 and 1873), and made the subject 
of close study by A. B. Kempe. To Sylvester is ascribed 


the general statement of the theory of contravariants, the 
discovery of the partial differential equations satisfied by 
the invariants and covariants of binary quantics, and the 
subject of mixed concomitants. In the American Journal of 
Mathematics are memoirs on binary and ternary quantics, 
elaborated partly with aid of F. Franklin, now professor at 
the Johns Hopkins University. At Oxford, Sylvester has 
opened up a new subject, the theory of reciprocants, treating 
of the functions of a dependent variable y and the functions 
of its differential coefficients in regard to x, which remain 
unaltered by the interchange of x and y. This theory is more 
general than one on differential invariants by Halphen (1878), 
and has been developed further by J. Hammond of Oxford, 
McMahon of Woolwich, A. R. Forsyth of Cambridge, and 
others. Sylvester playfully lays claim to the appellation of the 
Mathematical Adam, for the many names he has introduced 
into mathematics. Thus the terms invariant, discriminant, 
Hessian, Jacobian, are his. 

The great theory of invariants, developed in England 
mainly by Cayley and Sylvester, came to be studied earnestly 
in Germany, France, and Italy. One of the earliest in the 
field was Siegfried Heinrich Aronhold (1819-1884), who 
demonstrated the existence of invariants, S and T, of the 
ternary cubic. Hermite discovered evectants and the theorem 
of reciprocity named after him. Paul Gordan showed, with 
the aid of symbolic methods, that the number of distinct 
forms for a binary quantic is finite. Clebsch proved this to be 
true for quantics with any number of variables. A very much 


simpler proof of this was given in 1891, by David Hilbert of 
Konigsberg. In Italy, F. Brioschi of Milan and Fad de Bruno 
(1825-1888) contributed to the theory of invariants, the latter 
writing a text-book on binary forms, which ranks by the side of 
Salmon's treatise and those of Clebsch and Gordan. Among 
other writers on invariants are E. B. Christoffel, Wilhelm 
Fiedler, P. A. McMahon, J. W. L. Glaisher of Cambridge, 
Emory McClintock of New York. McMahon discovered that 
the theory of semi-invariants is a part of that of symmetric 
functions. The modern higher algebra has reached out and 
indissolubly connected itself with several other branches of 
mathematics — geometry, calculus of variations, mechanics. 
Clebsch extended the theory of binary forms to ternary, and 
applied the results to geometry. Clebsch, Klein, Weierstrass, 
Burckhardt, and Bianchi have used the theory of invariants in 
hyperelliptic and Abelian functions. 

In the theory of equations Lagrange, Argand, and Gauss 
furnished proof to the important theorem that every algebraic 
equation has a real or a complex root. Abel proved rigorously 
that the general algebraic equation of the fifth or of higher 
degrees cannot be solved by radicals (Crelle, I., 1826). A 
modification of Abel's proof was given by Wantzel. Before 
Abel, an Italian physician, Paolo Ruffini (1765-1822), had 
printed proofs of the insolvability, which were criticised by 
his countryman Malfatti. Though inconclusive, Ruffini's 
papers are remarkable as containing anticipations of Cauchy's 
theory of groups. [76] A transcendental solution of the quintic 
involving elliptic integrals was given by Hermite (Compt. 


Rend., 1858, 1865, 1866). After Hermite's first publication, 
Kronecker, in 1858, in a letter to Hermite, gave a second 
solution in which was obtained a simple resolvent of the sixth 
degree. Jerrard, inhis Mathematical Researches (1832-1835), 
reduced the quintic to the trinomial form by an extension of the 
method of Tschirnhausen. This important reduction had been 
effected as early as 1786 by E. S. Bring, a Swede, and brought 
out in a publication of the University of Lund. Jerrard, 
like Tschirnhausen, believed that his method furnished a 
general algebraic solution of equations of any degree. In 
1836 William R. Hamilton made a report on the validity of 
Jerrard's method, and showed that by his process the quintic 
could be transformed to any one of the four trinomial forms. 
Hamilton defined the limits of its applicability to higher 
equations. Sylvester investigated this question, What is the 
lowest degree an equation can have in order that it may admit 
of being deprived of i consecutive terms by aid of equations 
not higher than ith degree. He carried the investigation as 
far as i = 8, and was led to a series of numbers which he 
named "Hamilton's numbers." A transformation of equal 
importance to Jerrard's is that of Sylvester, who expressed 
the quintic as the sum of three fifth-powers. The covariants 
and invariants of higher equations have been studied much in 
recent years. 

Abel's proof that higher equations cannot always be solved 
algebraically led to the inquiry as to what equations of a given 
degree can be solved by radicals. Such equations are the ones 
discussed by Gauss in considering the division of the circle. 


Abel advanced one step further by proving that an irreducible 
equation can always be solved in radicals, if, of two of its 
roots, the one can be expressed rationally in terms of the 
other, provided that the degree of the equation is prime; if it is 
not prime, then the solution depends upon that of equations 
of lower degree. Through geometrical considerations, Hesse 
came upon algebraically solvable equations of the ninth 
degree, not included in the previous groups. The subject 
was powerfully advanced in Paris by the youthful Evariste 
Galois (born, 1811; killed in a duel, 1832), who introduced 
the notion of a group of substitutions. To him are due also 
some valuable results in relation to another set of equations, 
presenting themselves in the theory of elliptic functions, viz. 
the modular equations. Galois's labours gave birth to the 
important theory of substitutions, which has been greatly 
advanced by C. Jordan of Paris, J. A. Serret (1819-1885) of 
the Sorbonne in Paris, L. Kronecker (1823-1891) of Berlin, 
Klein of Gottingen, M. N other of Erlangen, C. Hermite 
of Paris, A. Capelli of Naples, L. Sylow of Friedrichshald, 
E. Netto of Giessen. Netto's book, the Substitutionstheorie, 
has been translated into English by F. N. Cole of the University 
of Michigan, who contributed to the theory. A simple group of 
504 substitutions of nine letters, discovered by Cole, has been 
shown by E. H. Moore of the University of Chicago to belong 
to a doubly-infinite system of simple groups. The theory 
of substitutions has important applications in the theory of 
differential equations. Kronecker published, in 1882, his 
Grundzilge einer Arithmetischen Theorie der Algebraischen 



Since Fourier and Budan, the solution of numerical equa- 
tions has been advanced by W. G. Horner of Bath, who 
gave an improved method of approximation [Philosophical 
Transactions, 1819). Jacques Charles Frangois Sturm 
(1803-1855), a native of Geneva, Switzerland, and the suc- 
cessor of Poisson in the chair of mechanics at the Sorbonne, 
published in 1829 his celebrated theorem determining the 
number and situation of roots of an equation comprised be- 
tween given limits. Sturm tells us that his theorem stared 
him in the face in the midst of some mechanical investigations 
connected with the motion of a compound pendulum. [77] 
This theorem, and Horner's method, offer together sure and 
ready means of finding the real roots of a numerical equation. 

The symmetric functions of the sums of powers of the roots 
of an equation, studied by Newton and Waring, was con- 
sidered more recently by Gauss, Cayley, Sylvester, Brioschi. 
Cayley gives rules for the "weight" and "order" of symmetric 

The theory of elimination was greatly advanced by Sylvester, 
Cayley, Salmon, Jacobi, Hesse, Cauchy, Brioschi, and Gordan. 
Sylvester gave the dialytic method [Philosophical Magazine, 
1840), and in 1852 established a theorem relating to the 
expression of an eliminant as a determinant. Cayley made 
a new statement of Bezout's method of elimination and 
established a general theory of elimination (1852). 



Under this head we find it convenient to consider the 
subjects of the differential and integral calculus, the calculus 
of variations, infinite series, probability, and differential 
equations. Prominent in the development of these subjects 
was Cauchy. 

Augustin-Louis Cauchy [78] (1789-1857) was born in 
Paris, and received his early education from his father. 
Lagrange and Laplace, with whom the father came in frequent 
contact, foretold the future greatness of the young boy. At the 
Ecole Centrale du Pantheon he excelled in ancient classical 
studies. In 1805 he entered the Polytechnic School, and two 
years later the Ecole des Ponts et Chaussees. Cauchy left 
for Cherbourg in 1810, in the capacity of engineer. Laplace's 
Mecanique Celeste and Lagrange's Fonctions Analytiques 
were among his book companions there. Considerations 
of health induced him to return to Paris after three years. 
Yielding to the persuasions of Lagrange and Laplace, he 
renounced engineering in favour of pure science. We find 
him next holding a professorship at the Polytechnic School. 
On the expulsion of Charles X., and the accession to the 
throne of Louis Philippe in 1830, Cauchy, being exceedingly 
conscientious, found himself unable to take the oath demanded 
of him. Being, in consequence, deprived of his positions, he 
went into voluntary exile. At Fribourg in Switzerland, Cauchy 
resumed his studies, and in 1831 was induced by the king 
of Piedmont to accept the chair of mathematical physics, 


especially created for him at the university of Turin. In 1833 
he obeyed the call of his exiled king, Charles X., to undertake 
the education of a grandson, the Duke of Bordeaux. This gave 
Cauchy an opportunity to visit various parts of Europe, and 
to learn how extensively his works were being read. Charles X. 
bestowed upon him the title of Baron. On his return to Paris 
in 1838, a chair in the College de France was offered to him, but 
the oath demanded of him prevented his acceptance. He was 
nominated member of the Bureau of Longitude, but declared 
ineligible by the ruling power. During the political events 
of 1848 the oath was suspended, and Cauchy at last became 
professor at the Polytechnic School. On the establishment of 
the second empire, the oath was re-instated, but Cauchy and 
Arago were exempt from it. Cauchy was a man of great piety, 
and in two of his publications staunchly defended the Jesuits. 

Cauchy was a prolific and profound mathematician. By a 
prompt publication of his results, and the preparation of stan- 
dard text-books, he exercised a more immediate and beneficial 
influence upon the great mass of mathematicians than any con- 
temporary writer. He was one of the leaders in infusing rigour 
into analysis. His researches extended over the field of series, of 
imaginaries, theory of numbers, differential equations, theory 
of substitutions, theory of functions, determinants, math- 
ematical astronomy, light, elasticity, etc., — covering pretty 
much the whole realm of mathematics, pure and applied. 

Encouraged by Laplace and Poisson, Cauchy published in 
1821 his Cours d 'Analyse de I'Ecole Roy ale Polytechnique, a 
work of great merit. Had it been studied more diligently by 


writers of text-books in England and the United States, many a 
lax and loose method of analysis hardly as yet eradicated from 
elementary text-books would have been discarded over half a 
century ago. Cauchy was the first to publish a rigorous proof 
of Taylor's theorem. He greatly improved the exposition of 
fundamental principles of the differential calculus by his mode 
of considering limits and his new theory on the continuity of 
functions. The method of Cauchy and Duhamel was accepted 
with favour by Hoiiel and others. In England special attention 
to the clear exposition of fundamental principles was given 
by De Morgan. Recent American treatises on the calculus 
introduce time as an independent variable, and the allied 
notions of velocity and acceleration — thus virtually returning 
to the method of fluxions. 

Cauchy made some researches on the calculus of variations. 
This subject is now in its essential principles the same as 
when it came from the hands of Lagrange. Recent studies 
pertain to the variation of a double integral when the limits 
are also variable, and to variations of multiple integrals in 
general. Memoirs were published by Gauss in 1829, Poisson 
in 1831, and Ostrogradsky of St. Petersburg in 1834, without, 
however, determining in a general manner the number and 
form of the equations which must subsist at the limits in 
case of a double or triple integral. In 1837 Jacobi published 
a memoir, showing that the difficult integrations demanded 
by the discussion of the second variation, by which the 
existence of a maximum or minimum can be ascertained, are 
included in the integrations of the first variation, and thus are 


superfluous. This important theorem, presented with great 
brevity by Jacobi, was elucidated and extended by V. A. 
Lebesgue, C. E. Delaunay, Eisenlohr, S. Spitzer, Hesse, and 
Clebsch. An important memoir by Sarrus on the question of 
determining the limiting equations which must be combined 
with the indefinite equations in order to determine completely 
the maxima and minima of multiple integrals, was awarded 
a prize by the French Academy in 1845, honourable mention 
being made of a paper by Delaunay. Sarrus's method was 
simplified by Cauchy. In 1852 G. Mainardi attempted to 
exhibit a new method of discriminating maxima and minima, 
and extended Jacobi's theorem to double integrals. Mainardi 
and F. Brioschi showed the value of determinants in exhibiting 
the terms of the second variation. In 1861 Isaac Todhunter 
(1820-1884) of St. John's College, Cambridge, published his 
valuable work on the History of the Progress of the Calculus of 
Variations, which contains researches of his own. In 1866 he 
published a most important research, developing the theory 
of discontinuous solutions (discussed in particular cases by 
Legendre), and doing for this subject what Sarrus had done 
for multiple integrals. 

The following are the more important authors of systematic 
treatises on the calculus of variations, and the dates of 
publication: Robert Woodhouse, Fellow of Caius College, 
Cambridge, 1810; Richard Abbatt in London, 1837; John 
Hewitt Jellett (1817-1888), once Provost of Trinity College, 
Dublin, 1850; G. W. Strauch in Zurich, 1849; Moigno and 
Lindelof, 1861; Lewis Buffett Carll of Flushing in New York, 



The lectures on definite integrals, delivered by Dirichlet in 
1858, have been elaborated into a standard work by G. F. 
Meyer. The subject has been treated most exhaustively by 
D. Bierens de Haan of Leiden in his Expose de la theorie des 
integrals definies, Amsterdam, 1862. 

The history of infinite series illustrates vividly the salient 
feature of the new era which analysis entered upon during 
the first quarter of this century. Newton and Leibniz felt 
the necessity of inquiring into the convergence of infinite 
series, but they had no proper criteria, excepting the test 
advanced by Leibniz for alternating series. By Euler and 
his contemporaries the formal treatment of series was greatly 
extended, while the necessity for determining the convergence 
was generally lost sight of. Euler reached some very pretty 
results on infinite series, now well known, and also some 
very absurd results, now quite forgotten. The faults of his 
time found their culmination in the Combinatorial School in 
Germany, which has now passed into deserved oblivion. At 
the beginning of the period now under consideration, the 
doubtful, or plainly absurd, results obtained from infinite 
series stimulated profounder inquiries into the validity of 
operations with them. Their actual contents came to be the 
primary, form a secondary, consideration. The first important 
and strictly rigorous investigation of series was made by Gauss 
in connection with the hypergeometric series. The criterion 
developed by him settles the question of convergence in every 
case which it is intended to cover, and thus bears the stamp of 


generality so characteristic of Gauss's writings. Owing to the 
strangeness of treatment and unusual rigour, Gauss's paper 
excited little interest among the mathematicians of that time. 

More fortunate in reaching the public was Cauchy, whose 
Analyse Algebrique of 1821 contains a rigorous treatment of 
series. All series whose sum does not approach a fixed limit as 
the number of terms increases indefinitely are called divergent. 
Like Gauss, he institutes comparisons with geometric series, 
and finds that series with positive terms are convergent or 
not, according as the nth root of the nth term, or the ratio 
of the (n + l)th term and the nth term, is ultimately less 
or greater than unity. To reach some of the cases where 
these expressions become ultimately unity and fail, Cauchy 
established two other tests. He showed that series with 
negative terms converge when the absolute values of the terms 
converge, and then deduces Leibniz's test for alternating 
series. The product of two convergent series was not found 
to be necessarily convergent. Cauchy's theorem that the 
product of two absolutely convergent series converges to the 
product of the sums of the two series was shown half a 
century later by F. Mertens of Graz to be still true if, of the 
two convergent series to be multiplied together, only one is 
absolutely convergent. 

The most outspoken critic of the old methods in series 
was Abel. His letter to his friend Holmboe (1826) contains 
severe criticisms. It is very interesting reading, even to 
modern students. In his demonstration of the binomial 
theorem he established the theorem that if two series and 


their product series are all convergent, then the product series 
will converge towards the product of the sums of the two 
given series. This remarkable result would dispose of the 
whole problem of multiplication of series if we had a universal 
practical criterion of convergency for semi- convergent series. 
Since we do not possess such a criterion, theorems have been 
recently established by A. Pringsheim of Munich and A. Voss 
of Wurzburg which remove in certain cases the necessity of 
applying tests of convergency to the product series by the 
application of tests to easier related expressions. Pringsheim 
reaches the following interesting conclusions: The product of 
two semi- convergent series can never converge absolutely, but 
a semi-convergent series, or even a divergent series, multiplied 
by an absolutely convergent series, may yield an absolutely 
convergent product. 

The researches of Abel and Cauchy caused a considerable 
stir. We are told that after a scientific meeting in which 
Cauchy had presented his first researches on series, Laplace 
hastened home and remained there in seclusion until he had 
examined the series in his Mecanique Celeste. Luckily, every 
one was found to be convergent! We must not conclude, 
however, that the new ideas at once displaced the old. On the 
contrary, the new views were generally accepted only after a 
severe and long struggle. As late as 1844 De Morgan began a 
paper on "divergent series" in this style: "I believe it will be 
generally admitted that the heading of this paper describes 
the only subject yet remaining, of an elementary character, 
on which a serious schism exists among mathematicians as to 


the absolute correctness or incorrectness of results." 

First in time in the evolution of more delicate criteria of 
convergence and divergence come the researches of Josef Lud- 
wig Raabe (Crelle, Vol. IX.); then follow those of De Morgan 
as given in his calculus. De Morgan established the loga- 
rithmic criteria which were discovered in part independently 
by J. Bertrand. The forms of these criteria, as given by 
Bertrand and by Ossian Bonnet, are more convenient than 
De Morgan's. It appears from Abel's posthumous papers that 
he had anticipated the above-named writers in establishing 
logarithmic criteria. It was the opinion of Bonnet that the 
logarithmic criteria never fail; but Du Bois-Reymond and 
Pringsheim have each discovered series demonstrably conver- 
gent in which these criteria fail to determine the convergence. 
The criteria thus far alluded to have been called by Pringsheim 
special criteria, because they all depend upon a comparison 
of the nth term of the series with special functions a n , n x , 
n(logn) x , etc. Among the first to suggest general criteria, 
and to consider the subject from a still wider point of view, 
culminating in a regular mathematical theory, was Kummer. 
He established a theorem yielding a test consisting of two 
parts, the first part of which was afterwards found to be 
superfluous. The study of general criteria was continued by 
U. Dini of Pisa, Paul Du Bois-Reymond, G. Kohn of Minden, 
and Pringsheim. Du Bois-Reymond divides criteria into two 
classes: criteria of the first kind and criteria of the second 
kind, according as the general nth term, or the ratio of the 
(n + l)th term and the nth term, is made the basis of research. 


Kummer's is a criterion of the second kind. A criterion of 
the first kind, analogous to this, was invented by Pringsheim. 
From the general criteria established by Du Bois-Reymond 
and Pringsheim respectively, all the special criteria can be 
derived. The theory of Pringsheim is very complete, and 
offers, in addition to the criteria of the first kind and second 
kind, entirely new criteria of a third kind, and also generalised 
criteria of the second kind, which apply, however, only to 
series with never increasing terms. Those of the third kind 
rest mainly on the consideration of the limit of the difference 
either of consecutive terms or of their reciprocals. In the 
generalised criteria of the second kind he does not consider the 
ratio of two consecutive terms, but the ratio of any two terms 
however far apart, and deduces, among others, two criteria 
previously given by Kohn and Ermakoff respectively. 

Difficult questions arose in the study of Fourier's series. [79] 
Cauchy was the first who felt the necessity of inquiring into 
its convergence. But his mode of proceeding was found 
by Dirichlet to be unsatisfactory. Dirichlet made the first 
thorough researches on this subject (Crelle, Vol. IV.). They 
culminate in the result that whenever the function does 
not become infinite, does not have an infinite number of 
discontinuities, and does not possess an infinite number of 
maxima and minima, then Fourier's series converges toward 
the value of that function at all places, except points of 
discontinuity, and there it converges toward the mean of the 
two boundary values. Schlafli of Bern and Du Bois-Reymond 
expressed doubts as to the correctness of the mean value, which 


were, however, not well founded. Dirichlet's conditions are 
sufficient, but not necessary. Lipschitz, of Bonn, proved that 
Fourier's series still represents the function when the number 
of discontinuities is infinite, and established a condition on 
which it represents a function having an infinite number of 
maxima and minima. Dirichlet's belief that all continuous 
functions can be represented by Fourier's series at all points 
was shared by Riemann and H. Hankel, but was proved to be 
false by Du Bois-Reymond and H. A. Schwarz. 

Riemann inquired what properties a function must have, so 
that there may be a trigonometric series which, whenever it 
is convergent, converges toward the value of the function. He 
found necessary and sufficient conditions for this. They do 
not decide, however, whether such a series actually represents 
the function or not. Riemann rejected Cauchy's definition of 
a definite integral on account of its arbitrariness, gave a new 
definition, and then inquired when a function has an integral. 
His researches brought to light the fact that continuous 
functions need not always have a differential coefficient. But 
this property, which was shown by Weierstrass to belong to 
large classes of functions, was not found necessarily to exclude 
them from being represented by Fourier's series. Doubts on 
some of the conclusions about Fourier's series were thrown 
by the observation, made by Weierstrass, that the integral of 
an infinite series can be shown to be equal to the sum of the 
integrals of the separate terms only when the series converges 
uniformly within the region in question. The subject of 
uniform convergence was investigated by Philipp Ludwig 


Seidel (1848) and G. G. Stokes (1847), and has assumed great 
importance in Weierstrass' theory of functions. It became 
necessary to prove that a trigonometric series representing 
a continuous function converges uniformly. This was done 
by Heinrich Eduard Heine (1821-1881), of Halle. Later 
researches on Fourier's series were made by G. Cantor and 
Du Bois-Reymond. 

As compared with the vast development of other mathe- 
matical branches, the theory of probability has made very 
insignificant progress since the time of Laplace. Improvements 
and simplifications in the mode of exposition have been made 
by A. De Morgan, G. Boole, A. Meyer (edited by E. Czuber), 
J. Bertrand. Cournot's and Westergaard's treatment of insur- 
ance and the theory of life-tables are classical. Applications of 
the calculus to statistics have been made by L. A. J. Quetelet 
(1796-1874), director of the observatory at Brussels; by Lexis; 
Harald Westergaard, of Copenhagen; and Dusing. 

Worthy of note is the rejection of inverse probability by 

the best authorities of our time. This branch of probability 

had been worked out by Thomas Bayes (died 1761) and by 

Laplace (Bk. II., Ch. VI. of his Theorie Analytique). By it 

some logicians have explained induction. For example, if a 

man, who has never heard of the tides, were to go to the shore 

of the Atlantic Ocean and witness on m successive days the 

rise of the sea, then, says Quetelet, he would be entitled to 

vn -\- 1 
conclude that there was a probability equal to that 

m + 2 

the sea would rise next day. Putting m = 0, it is seen that 
this view rests upon the unwarrantable assumption that the 


probability of a totally unknown event is \, or that of all 
theories proposed for investigation one-half are true. W. S. 
Jevons in his Principles of Science founds induction upon 
the theory of inverse probability, and F. Y. Edgeworth also 
accepts it in his Mathematical Psychics. 

The only noteworthy recent addition to probability is the 
subject of "local probability," developed by several English 
and a few American and French mathematicians. The earliest 
problem on this subject dates back to the time of Buffon, 
the naturalist, who proposed the problem, solved by himself 
and Laplace, to determine the probability that a short needle, 
thrown at random upon a floor ruled with equidistant parallel 
lines, will fall on one of the lines. Then came Sylvester's four- 
point problem: to find the probability that four points, taken 
at random within a given boundary, shall form a re-entrant 
quadrilateral. Local probability has been studied in England 
by A. R. Clarke, H. McColl, S. Watson, J. Wolstenholme, but 
with greatest success by M. W. Crofton of the military school 
at Woolwich. It was pursued in America by E. B. Seitz; in 
France by C. Jordan, E. Lemoine, E. Barbier, and others. 
Through considerations of local probability, Crofton was led 
to the evaluation of certain definite integrals. 

The first full scientific treatment of differential equations was 
given by Lagrange and Laplace. This remark is especially true 
of partial differential equations. The latter were investigated 
in more recent time by Monge, Pfaff, Jacobi, Emile Bour 
(1831-1866) of Paris, A. Weiler, Clebsch, A. N. Korkine of 
St. Petersburg, G. Boole, A. Meyer, Cauchy, Serret, Sophus 


Lie, and others. In 1873 their researches, on partial differential 
equations of the first order, were presented in text-book form 
by Paul Mansion, of the University of Gand. The keen 
researches of Johann Friedrich Pfaff (1795-1825) marked 
a decided advance. He was an intimate friend of young 
Gauss at Gottingen. Afterwards he was with the astronomer 
Bode. Later he became professor at Helmstadt, then at Halle. 
By a peculiar method, Pfaff found the general integration 
of partial differential equations of the first order for any 
number of variables. Starting from the theory of ordinary 
differential equations of the first order in n variables, he 
gives first their general integration, and then considers the 
integration of the partial differential equations as a particular 
case of the former, assuming, however, as known, the general 
integration of differential equations of any order between 
two variables. His researches led Jacobi to introduce the 
name "Pfaffian problem." From the connection, observed by 
Hamilton, between a system of ordinary differential equations 
(in analytical mechanics) and a partial differential equation, 
Jacobi drew the conclusion that, of the series of systems whose 
successive integration Pfaff's method demanded, all but the 
first system were entirely superfluous. Clebsch considered 
Pfaff's problem from a new point of view, and reduced it to 
systems of simultaneous linear partial differential equations, 
which can be established independently of each other without 
any integration. Jacobi materially advanced the theory of 
differential equations of the first order. The problem to 
determine unknown functions in such a way that an integral 


containing these functions and their differential coefficients, 
in a prescribed manner, shall reach a maximum or minimum 
value, demands, in the first place, the vanishing of the first 
variation of the integral. This condition leads to differential 
equations, the integration of which determines the functions. 
To ascertain whether the value is a maximum or a minimum, 
the second variation must be examined. This leads to new 
and difficult differential equations, the integration of which, 
for the simpler cases, was ingeniously deduced by Jacobi 
from the integration of the differential equations of the first 
variation. Jacobi's solution was perfected by Hesse, while 
Clebsch extended to the general case Jacobi's results on 
the second variation. Cauchy gave a method of solving 
partial differential equations of the first order having any 
number of variables, which was corrected and extended by 
Serret, J. Bertrand, O. Bonnet in France, and Imschenetzky 
in Russia. Fundamental is the proposition of Cauchy that 
every ordinary differential equation admits in the vicinity 
of any non-singular point of an integral, which is synectic 
within a certain circle of convergence, and is developable 
by Taylor's theorem. Allied to the point of view indicated 
by this theorem is that of Riemann, who regards a function 
of a single variable as defined by the position and nature 
of its singularities, and who has applied this conception to 
that linear differential equation of the second order, which 
is satisfied by the hypergeometric series. This equation was 
studied also by Gauss and Kummer. Its general theory, 
when no restriction is imposed upon the value of the variable, 


has been considered by J. Tannery, of Paris, who employed 
Fuchs' method of linear differential equations and found all of 
Rummer's twenty-four integrals of this equation. This study 
has been continued by Edouard Goursat of Paris. 

A standard text-book on Differential Equations, including 
original matter on integrating factors, singular solutions, and 
especially on symbolical methods, was prepared in 1859 by 
George Boole (1815-1864), at one time professor in Queen's 
University, Cork, Ireland. He was a native of Lincoln, and a 
self-educated mathematician of great power. His treatise on 
Finite Differences (1860) and his Laws of Thought (1854) are 
works of high merit. 

The fertility of the conceptions of Cauchy and Riemann with 
regard to differential equations is attested by the researches 
to which they have given rise on the part of Lazarus Fuchs 
of Berlin (born 1835), Felix Klein of Gottingen (born 1849), 
Henri Poincare of Paris (born 1854), and others. The 
study of linear differential equations entered a new period 
with the publication of Fuchs' memoirs of 1866 and 1868. 
Before this, linear equations with constant coefficients were 
almost the only ones for which general methods of integration 
were known. While the general theory of these equations 
has recently been presented in a new light by Hermite, 
Darboux, and Jordan, Fuchs began the study from the more 
general standpoint of the linear differential equations whose 
coefficients are not constant. He directed his attention mainly 
to those whose integrals are all regular. If the variable be made 
to describe all possible paths enclosing one or more of the 


critical points of the equation, we have a certain substitution 
corresponding to each of the paths; the aggregate of all these 
substitutions being called a group. The forms of integrals of 
such equations were examined by Fuchs and by G. Frobenius 
by independent methods. Logarithms generally appear in the 
integrals of a group, and Fuchs and Frobenius investigated the 
conditions under which no logarithms shall appear. Through 
the study of groups the reducibility or irreducibility of linear 
differential equations has been examined by Frobenius and 
Leo Konigsberger. The subject of linear differential equations, 
not all of whose integrals are regular, has been attacked by 
G. Frobenius of Berlin, W. Thome of Greifswald (born 1841), 
and Poincare, but the resulting theory of irregular integrals is 
as yet in very incomplete form. 

The theory of invariants associated with linear differential 
equations has been developed by Halphen and by A. R. 

The researches above referred to are closely connected with 
the theory of functions and of groups. Endeavours have thus 
been made to determine the nature of the function defined by 
a differential equation from the differential equation itself, and 
not from any analytical expression of the function, obtained 
first by solving the differential equation. Instead of studying 
the properties of the integrals of a differential equation for 
all the values of the variable, investigators at first contented 
themselves with the study of the properties in the vicinity of a 
given point. The nature of the integrals at singular points and 
at ordinary points is entirely different. Albert Briot (1817- 


1882) and Jean Claude Bouquet (1819-1885), both of Paris, 

studied the case when, near a singular point, the differential 

equations take the form (x — xq) — = / (xy). Fuchs gave the 

ax j 

development in series of the integrals for the particular case 
of linear equations. Poincare did the same for the case when 
the equations are not linear, as also for partial differential 
equations of the first order. The developments for ordinary 
points were given by Cauchy and Madame Kowalevsky. 

The attempt to express the integrals by developments that 
are always convergent and not limited to particular points in 
a plane necessitates the introduction of new transcendents, 
for the old functions permit the integration of only a small 
number of differential equations. Poincare tried this plan with 
linear equations, which were then the best known, having 
been studied in the vicinity of given points by Fuchs, Thome, 
Frobenius, Schwarz, Klein, and Halphen. Confining himself 
to those with rational algebraical coefficients, Poincare was 
able to integrate them by the use of functions named by him 
Fuchsians. [81] He divided these equations into "families." 
If the integral of such an equation be subjected to a certain 
transformation, the result will be the integral of an equation 
belonging to the same family. The new transcendents have 
a great analogy to elliptic functions; while the region of the 
latter may be divided into parallelograms, each representing a 
group, the former may be divided into curvilinear polygons, so 
that the knowledge of the function inside of one polygon carries 
with it the knowledge of it inside the others. Thus Poincare 
arrives at what he calls Fuchsian groups. He found, moreover, 


that Fuchsian functions can be expressed as the ratio of two 
transcendents (theta-fuchsians) in the same way that elliptic 
functions can be. If, instead of linear substitutions with 
real coefficients, as employed in the above groups, imaginary 
coefficients be used, then discontinuous groups are obtained, 
which he called Kleinians. The extension to non-linear 
equations of the method thus applied to linear equations has 
been begun by Fuchs and Poincare. 

We have seen that among the earliest of the several kinds 
of "groups" are the finite discontinuous groups (groups in 
the theory of substitution), which since the time of Galois 
have become the leading concept in the theory of algebraic 
equations; that since 1876 Felix Klein, H. Poincare, and others 
have applied the theory of finite and infinite discontinuous 
groups to the theory of functions and of differential equations. 
The finite continuous groups were first made the subject of 
general research in 1873 by Sophus Lie, now of Leipzig, and 
applied by him to the integration of ordinary linear partial 
differential equations. 

Much interest attaches to the determination of those linear 
differential equations which can be integrated by simpler 
functions, such as algebraic, elliptic, or Abelian. This has 
been studied by C . Jordan, P. Appel of Paris (born 1858), and 

The mode of integration above referred to, which makes 
known the properties of equations from the standpoint of 
the theory of functions, does not suffice in the application 
of differential equations to questions of mechanics. If we 


consider the function as denning a plane curve, then the 
general form of the curve does not appear from the above 
mode of investigation. It is, however, often desirable to 
construct the curves defined by differential equations. Studies 
having this end in view have been carried on by Briot and 
Bouquet, and by Poincare. [81] 

The subject of singular solutions of differential equations 
has been materially advanced since the time of Boole by 
G. Darboux and Cayley. The papers prepared by these 
mathematicians point out a difficulty as yet unsurmounted: 
whereas a singular solution, from the point of view of the 
integrated equation, ought to be a phenomenon of universal, 
or at least of general occurrence, it is, on the other hand, a 
very special and exceptional phenomenon from the point of 
view of the differential equation. [89] A geometrical theory 
of singular solutions resembling the one used by Cayley was 
previously employed by W. W. Johnson of Annapolis. 

An advanced Treatise on Linear Differential Equations 
(1889) was brought out by Thomas Craig of the Johns Hopkins 
University. He chose the algebraic method of presentation 
followed by Hermite and Poincare, instead of the geometric 
method preferred by Klein and Schwarz. A notable work, the 
Traite d' Analyse, is now being published by Emile Picard of 
Paris, the interest of which is made to centre in the subject of 
differential equations. 



We begin our sketch of the vast progress in the theory 
of functions by considering the special class called elliptic 
functions. These were richly developed by Abel and Jacobi. 

Niels Henrick Abel (1802-1829) was born at Findoe in 
Norway, and was prepared for the university at the cathedral 
school in Christiania. He exhibited no interest in mathematics 
until 1818, when B. Holmboe became lecturer there, and 
aroused Abel's interest by assigning original problems to the 
class. Like Jacobi and many other young men who became 
eminent mathematicians, Abel found the first exercise of his 
talent in the attempt to solve by algebra the general equation 
of the fifth degree. In 1821 he entered the University in 
Christiania. The works of Euler, Lagrange, and Legendre 
were closely studied by him. The idea of the inversion of 
elliptic functions dates back to this time. His extraordinary 
success in mathematical study led to the offer of a stipend 
by the government, that he might continue his studies in 
Germany and France. Leaving Norway in 1825, Abel visited 
the astronomer, Schumacher, in Hamburg, and spent six 
months in Berlin, where he became intimate with August 
Leopold Crelle (1780-1855), and met Steiner. Encouraged 
by Abel and Steiner, Crelle started his journal in 1826. Abel 
began to put some of his work in shape for print. His 
proof of the impossibility of solving the general equation of 
the fifth degree by radicals, — first printed in 1824 in a very 
concise form, and difficult of apprehension, — was elaborated 


in greater detail, and published in the first volume. He 
entered also upon the subject of infinite series (particularly 
the binomial theorem, of which he gave in Crelle 's Journal a 
rigid general investigation), the study of functions, and of the 
integral calculus. The obscurities everywhere encountered 
by him owing to the prevailing loose methods of analysis he 
endeavoured to clear up. For a short time he left Berlin 
for Freiberg, where he had fewer interruptions to work, and 
it was there that he made researches on hyperelliptic and 
Abelian functions. In July, 1826, Abel left Germany for 
Paris without having met Gauss! Abel had sent to Gauss his 
proof of 1824 of the impossibility of solving equations of the 
fifth degree, to which Gauss never paid any attention. This 
slight, and a haughtiness of spirit which he associated with 
Gauss, prevented the genial Abel from going to Gottingen. A 
similar feeling was entertained by him later against Cauchy. 
Abel remained ten months in Paris. He met there Dirichlet, 
Legendre, Cauchy, and others; but was little appreciated. He 
had already published several important memoirs in Crelle 's 
Journal, but by the French this new periodical was as yet 
hardly known to exist, and Abel was too modest to speak 
of his own work. Pecuniary embarrassments induced him to 
return home after a second short stay in Berlin. At Christiania 
he for some time gave private lessons, and served as docent. 
Crelle secured at last an appointment for him at Berlin; but 
the news of it did not reach Norway until after the death of 
Abel at Froland. [82] 

At nearly the same time with Abel, Jacobi published articles 


on elliptic functions. Legendre's favourite subject, so long 
neglected, was at last to be enriched by some extraordinary 
discoveries. The advantage to be derived by inverting the 
elliptic integral of the first kind and treating it as a function 
of its amplitude (now called elliptic function) was recognised 
by Abel, and a few months later also by Jacobi. A second 
fruitful idea, also arrived at independently by both, is the 
introduction of imaginaries leading to the observation that 
the new functions simulated at once trigonometric and expo- 
nential functions. For it was shown that while trigonometric 
functions had only a real period, and exponential only an 
imaginary, elliptic functions had both sorts of periods. These 
two discoveries were the foundations upon which Abel and 
Jacobi, each in his own way, erected beautiful new structures. 
Abel developed the curious expressions representing elliptic 
functions by infinite series or quotients of infinite products. 
Great as were the achievements of Abel in elliptic functions, 
they were eclipsed by his researches on what are now called 
Abelian functions. Abel's theorem on these functions was 
given by him in several forms, the most general of these being 
that in his Memoire sur une propriete generate d'une classe 
tres-etendue de fonctions transcendentes (1826). The history 
of this memoir is interesting. A few months after his arrival 
in Paris, Abel submitted it to the French Academy. Cauchy 
and Legendre were appointed to examine it; but said nothing 
about it until after Abel's death. In a brief statement of the 
discoveries in question, published by Abel in Crelle 's Journal, 
1829, reference is made to that memoir. This led Jacobi to 


inquire of Legendre what had become of it. Legendre says 
that the manuscript was so badly written as to be illegible, 
and that Abel was asked to hand in a better copy, which he 
neglected to do. The memoir remained in Cauchy's hands. 
It was not published until 1841. By a singular mishap, the 
manuscript was lost before the proof-sheets were read. 

In its form, the contents of the memoir belongs to the 
integral calculus. Abelian integrals depend upon an irrational 
function y which is connected with x by an algebraic equation 
F(x,y) = 0. Abel's theorem asserts that a sum of such 
integrals can be expressed by a definite number p of similar 
integrals, where p depends merely on the properties of the 
equation F(x,y) = 0. It was shown later that p is the 
deficiency of the curve F(x,y) = 0. The addition theorems 
of elliptic integrals are deducible from Abel's theorem. The 
hyperelliptic integrals introduced by Abel, and proved by him 
to possess multiple periodicity, are special cases of Abelian 
integrals whenever p = or > 3. The reduction of Abelian to 
elliptic integrals has been studied mainly by Jacobi, Hermite, 
Konigsberger, Brioschi, Goursat, E. Picard, and O. Bolza of 
the University of Chicago. 

Two editions of Abel's works have been published: the first 
by Holmboe in 1839, and the second by Sylow and Lie in 1881. 

Abel's theorem was pronounced by Jacobi the greatest 
discovery of our century on the integral calculus. The 
aged Legendre, who greatly admired Abel's genius, called it 
"monumentum aere perennius." During the few years of work 
allotted to the young Norwegian, he penetrated new fields of 


research, the development of which has kept mathematicians 
busy for over half a century. 

Some of the discoveries of Abel and Jacobi were anticipated 
by Gauss. In the Disquisitiones Arithmeticce he observed that 
the principles which he used in the division of the circle were 
applicable to many other functions, besides the circular, and 

particularly to the transcendents dependent on the integral 


. From this Jacobi [83] concluded that Gauss had 

thirty years earlier considered the nature and properties of 
elliptic functions and had discovered their double periodicity. 
The papers in the collected works of Gauss confirm this 

Carl Gustav Jacob Jacobi [84] (1804-1851) was born of 
Jewish parents at Potsdam. Like many other mathematicians 
he was initiated into mathematics by reading Euler. At 
the University of Berlin, where he pursued his mathematical 
studies independently of the lecture courses, he took the 
degree of Ph.D. in 1825. After giving lectures in Berlin for two 
years, he was elected extraordinary professor at Konigsberg, 
and two years later to the ordinary professorship there. After 
the publication of his Fundamenta Nova he spent some time 
in travel, meeting Gauss in Gottingen, and Legendre, Fourier, 
Poisson, in Paris. In 1842 he and his colleague, Bessel, 
attended the meetings of the British Association, where they 
made the acquaintance of English mathematicians. 

His early researches were on Gauss' approximation to 
the value of definite integrals, partial differential equations, 
Legendre's coefficients, and cubic residues. He read Legendre's 


Exercises, which give an account of elliptic integrals. When 
he returned the book to the library, he was depressed in spirits 
and said that important books generally excited in him new 
ideas, but that this time he had not been led to a single original 
thought. Though slow at first, his ideas flowed all the richer 
afterwards. Many of his discoveries in elliptic functions were 
made independently by Abel. Jacobi communicated his first 
researches to Crelle's Journal. In 1829, at the age of twenty- 
five, he published his Fundamenta Nova Theorize Functionum 
Ellipticarum, which contains in condensed form the main 
results in elliptic functions. This work at once secured for 
him a wide reputation. He then made a closer study of 
theta-functions and lectured to his pupils on a new theory of 
elliptic functions based on the theta-functions. He developed 
a theory of transformation which led him to a multitude 
of formulae containing q, a transcendental function of the 
modulus, defined by the equation q = e~ 7rk / k . He was also led 
by it to consider the two new functions H and O, which taken 
each separately with two different arguments are the four 
(single) theta-functions designated by the 61, 62, 63, 64. [56] 
In a short but very important memoir of 1832, he shows that 
for the hyperelliptic integral of any class the direct functions 
to which Abel's theorem has reference are not functions of a 
single variable, such as the elliptic sn, en, dn, but functions of 
p variables. [56] Thus in the case p = 2, which Jacobi especially 
considers, it is shown that Abel's theorem has reference to two 
functions X(u,v), Xi(u,v), each of two variables, and gives in 
effect an addition-theorem for the expression of the functions 


X(u + u', v + v'), X\(u + u', v + i/) algebraically in terms of the 
functions X(u, v), Xi(u,v), X(u',v'), Xi(u',v'). By the memoirs 
of Abel and Jacobi it may be considered that the notion of 
the Abelian function of p variables was established and the 
addition-theorem for these functions given. Recent studies 
touching Abelian functions have been made by Weierstrass, 
E. Picard, Madame Kowalevski, and Poincare. Jacobi's work 
on differential equations, determinants, dynamics, and the 
theory of numbers is mentioned elsewhere. 

In 1842 Jacobi visited Italy for a few months to recuperate 
his health. At this time the Prussian government gave him a 
pension, and he moved to Berlin, where the last years of his 
life were spent. 

The researches on functions mentioned thus far have been 
greatly extended. In 1858 Charles Hermite of Paris (born 
1822), introduced in place of the variable q of Jacobi a new 
variable uj connected with it by the equation q = e inuj , so that 
to = ik! /k, and was led to consider the functions 4>(uj), tp{uj), 
x(w). [56] Henry Smith regarded a theta- function with the 
argument equal to zero, as a function of uj. This he called an 
omega- function, while the three functions </>(w), tp(oj), x( w )j 
are his modular functions. Researches on theta-functions 
with respect to real and imaginary arguments have been made 
by Meissel of Kiel, J. Thomae of Jena, Alfred Enneper of 
Gottingen (1830-1885). A general formula for the product 
of two theta-functions was given in 1854 by H. Schroter of 
Breslau (1829-1892). These functions have been studied 
also by Cauchy, Konigsberger of Heidelberg (born 1837), 


F. S. Richelot of Konigsberg (1808-1875), Johann Georg 
Rosenhain of Konigsberg (1816-1887), L. Schlafii of Bern 
(born 1818). [85] 

Legendre's method of reducing an elliptic differential to 
its normal form has called forth many investigations, most 
important of which are those of Richelot and of Weierstrass of 

The algebraic transformations of elliptic functions involve 
a relation between the old modulus and the new one which 
Jacobi expressed by a differential equation of the third order, 
and also by an algebraic equation, called by him "modular 
equation." The notion of modular equations was familiar to 
Abel, but the development of this subject devolved upon later 
investigators. These equations have become of importance in 
the theory of algebraic equations, and have been studied by 
Sohnke, E. Mathieu, L. Konigsberger, E. Betti of Pisa (died 
1892), C. Hermite of Paris, Joubert of Angers, Francesco 
Brioschi of Milan, Schlafii, H. Schroter, M. Gudermann of 
Cleve, Giitzlaff. 

Felix Klein of Gottingen has made an extensive study of 
modular functions, dealing with a type of operations lying 
between the two extreme types, known as the theory of 
substitutions and the theory of invariants and covariants. 
Klein's theory has been presented in book-form by his pupil, 
Robert Fricke. The bolder features of it were first published 
in his Ikosaeder, 1884. His researches embrace the theory of 
modular functions as a specific class of elliptic functions, the 
statement of a more general problem as based on the doctrine 


of groups of operations, and the further development of the 
subject in connection with a class of Riemann's surfaces. 

The elliptic functions were expressed by Abel as quotients 
of doubly infinite products. He did not, however, inquire 
rigorously into the convergency of the products. In 1845 
Cayley studied these products, and found for them a complete 
theory, based in part upon geometrical interpretation, which 
he made the basis of the whole theory of elliptic functions. 
Eisenstein discussed by purely analytical methods the general 
doubly infinite product, and arrived at results which have been 
greatly simplified in form by the theory of primary factors, 
due to Weierstrass. A certain function involving a doubly 
infinite product has been called by Weierstrass the sigma- 
function, and is the basis of his beautiful theory of elliptic 
functions. The first systematic presentation of Weierstrass' 
theory of elliptic functions was published in 1886 by G. H. 
Halphen in his Theorie des fonctions elliptiques et des leurs 
applications. Applications of these functions have been given 
also by A. G. Greenhill. Generalisations analogous to those 
of Weierstrass on elliptic functions have been made by Felix 
Klein on hyperelliptic functions. 

Standard works on elliptic functions have been published by 
Briot and Bouquet (1859), by Konigsberger, Cayley, Heinrich 
Durege of Prague (1821-1893), and others. 

Jacobi's work on Abelian and theta-functions was greatly 
extended by Adolph Gopel (1812-1847), professor in a 
gymnasium near Potsdam, and Johann Georg Rosenhain 
of Konigsberg (1816-1887). Gopel in his Theorice transcen- 


dentium primi ordinis adumbratio levis (Crelle, 35, 1847) and 
Rosenhain in several memoirs established each independently, 
on the analogy of the single theta-functions, the functions of 
two variables, called double theta-functions, and worked out 
in connection with them the theory of the Abelian functions of 
two variables. The theta-relations established by Gopel and 
Rosenhain received for thirty years no further development, 
notwithstanding the fact that the double theta series came 
to be of increasing importance in analytical, geometrical, and 
mechanical problems, and that Hermite and Konigsberger 
had considered the subject of transformation. Finally, the 
investigations of C. W. Borchardt of Berlin (1817-1880), 
treating of the representation of Kummer's surface by Gopel's 
biquadratic relation between four theta-functions of two vari- 
ables, and researches of H. H. Weber of Marburg, F. Prym 
of Wiirzburg, Adolf Krazer, and Martin Krause of Dresden 
led to broader views. Researches on double theta-functions, 
made by Cayley, were extended to quadruple theta-functions 
by Thomas Craig of the Johns Hopkins University. 

Starting with the integrals of the most general form and 
considering the inverse functions corresponding to these 
integrals (the Abelian functions of p variables), Riemann 
defined the theta-functions of p variables as the sum of 
a p-tuply infinite series of exponentials, the general term 
depending on p variables. Riemann shows that the Abelian 
functions are algebraically connected with theta-functions of 
the proper arguments, and presents the theory in the broadest 
form. [56] He rests the theory of the multiple theta-functions 


upon the general principles of the theory of functions of a 
complex variable. 

Through the researches of A. Brill of Tubingen, M. Nother 
of Erlangen, and Ferdinand Lindemann of Munich, made in 
connection with Riemann-Roch's theorem and the theory of 
residuation, there has grown out of the theory of Abelian 
functions a theory of algebraic functions and point-groups on 
algebraic curves. 

Before proceeding to the general theory of functions, we 
make mention of the "calculus of functions," studied chiefly 
by C. Babbage, J. F. W. Herschel, and De Morgan, which was 
not so much a theory of functions as a theory of the solution of 
functional equations by means of known functions or symbols. 

The history of the general theory of functions begins with 
the adoption of new definitions of a function. With the 
Bernoullis and Leibniz, y was called a function of x, if there 
existed an equation between these variables which made it 
possible to calculate y for any given value of x lying anywhere 
between — oo and +00. The study of Fourier's theory of heat 
led Dirichlet to a new definition: y is called a function of x, 
if y possess one or more definite values for each of certain 
values that x is assumed to take in an interval xq to x\. In 
functions thus defined, there need be no analytical connection 
between y and x, and it becomes necessary to look for possible 
discontinuities. A great revolution in the ideas of a function 
was brought about by Cauchy when, in a function as defined 
by Dirichlet, he gave the variables imaginary values, and when 
he extended the notion of a definite integral by letting the 


variable pass from one limit to the other by a succession of 
imaginary values along arbitrary paths. Cauchy established 
several fundamental theorems, and gave the first great impulse 
to the study of the general theory of functions. His researches 
were continued in France by Puiseux and Liouville. But more 
profound investigations were made in Germany by Riemann. 
Georg Friedrich Bernhard Riemann (1826 1866) was 
born at Breselenz in Hanover. His father wished him to 
study theology, and he accordingly entered upon philological 
and theological studies at Gottingen. He attended also some 
lectures on mathematics. Such was his predilection for this 
science that he abandoned theology. After studying for a time 
under Gauss and Stern, he was drawn, in 1847, to Berlin by a 
galaxy of mathematicians, in which shone Dirichlet, Jacobi, 
Steiner, and Eisenstein. Returning to Gottingen in 1850, 
he studied physics under Weber, and obtained the doctorate 
the following year. The thesis presented on that occasion, 
Grundlagen fur eine allgemeine Theorie der Funktionen einer 
verdnderlichen complexen Grosse, excited the admiration 
of Gauss to a very unusual degree, as did also Riemann's 
trial lecture, Ueber die Hypothesen welche der Geometrie zu 
Grunde liegen. Riemann's Habilitationsschrift was on the 
Representation of a Function by means of a Trigonometric 
Series, in which he advanced materially beyond the position of 
Dirichlet. Our hearts are drawn to this extraordinarily gifted 
but shy genius when we read of the timidity and nervousness 
displayed when he began to lecture at Gottingen, and of 
his jubilation over the unexpectedly large audience of eight 


students at his first lecture on differential equations. 

Later he lectured on Abelian functions to a class of three 
only, — Schering, Bjerknes, and Dedekind. Gauss died in 1855, 
and was succeeded by Dirichlet. On the death of the latter, 
in 1859, Riemann was made ordinary professor. In 1860 
he visited Paris, where he made the acquaintance of French 
mathematicians. The delicate state of his health induced him 
to go to Italy three times. He died on his last trip at Selasca, 
and was buried at Biganzolo. 

Like all of Riemann's researches, those on functions were 
profound and far-reaching. He laid the foundation for a 
general theory of functions of a complex variable. The 
theory of potential, which up to that time had been used 
only in mathematical physics, was applied by him in pure 

mathematics. He accordingly based his theory of functions on 

d ii u 
the partial differential equation, — — k + tt^t = Au = 0, which 

ox 1 ay 1 

must hold for the analytical function w = u + ivoiz = x + iy. It 
had been proved by Dirichlet that (for a plane) there is always 
one, and only one, function of x and y, which satisfies An = 0, 
and which, together with its differential quotients of the first 
two orders, is for all values of x and y within a given area 
one-valued and continuous, and which has for points on the 
boundary of the area arbitrarily given values. [86] Riemann 
called this "Dirichlet's principle," but the same theorem 
was stated by Green and proved analytically by Sir William 
Thomson. It follows then that w is uniquely determined for 
all points within a closed surface, if u is arbitrarily given for all 


points on the curve, whilst v is given for one point within the 
curve. In order to treat the more complicated case where w has 
n values for one value of z, and to observe the conditions about 
continuity, Riemann invented the celebrated surfaces, known 
as "Riemann's surfaces," consisting of n coincident planes 
or sheets, such that the passage from one sheet to another 
is made at the branch-points, and that the n sheets form 
together a multiply-connected surface, which can be dissected 
by cross-cuts into a singly-connected surface. The n-valued 
function w becomes thus a one-valued function. Aided by 
researches of J. Liiroth of Freiburg and of Clebsch, W. K. 
Clifford brought Riemann's surface for algebraic functions to 
a canonical form, in which only the two last of the n leaves are 
multiply-connected, and then transformed the surface into the 
surface of a solid with p holes. A. Hurwitz of Zurich discussed 
the question, how far a Riemann's surface is determinate by 
the assignment of its number of sheets, its branch-points and 
branch-lines. [62] 

Riemann's theory ascertains the criteria which will deter- 
mine an analytical function by aid of its discontinuities and 
boundary conditions, and thus defines a function indepen- 
dently of a mathematical expression. In order to show that 
two different expressions are identical, it is not necessary to 
transform one into the other, but it is sufficient to prove the 
agreement to a far less extent, merely in certain critical points. 

Riemann's theory, as based on Dirichlet's principle (Thom- 
son's theorem), is not free from objections. It has become 
evident that the existence of a derived function is not a con- 


sequence of continuity, and that a function may be integrable 
without being differentiable. It is not known how far the meth- 
ods of the infinitesimal calculus and the calculus of variations 
(by which Dirichlet's principle is established) can be applied 
to an unknown analytical function in its generality. Hence the 
use of these methods will endow the functions with properties 
which themselves require proof. Objections of this kind to 
Riemann's theory have been raised by Kronecker, Weierstrass, 
and others, and it has become doubtful whether his most im- 
portant theorems are actually proved. In consequence of this, 
attempts have been made to graft Riemann's speculations 
on the more strongly rooted methods of Weierstrass. The 
latter developed a theory of functions by starting, not with 
the theory of potential, but with analytical expressions and 
operations. Both applied their theories to Abelian functions, 
but there Riemann's work is more general. [86] 

The theory of functions of one complex variable has been 
studied since Riemann's time mainly by Karl Weierstrass 
of Berlin (born 1815), Gustaf Mittag-Leffler of Stockholm 
(born 1846), and Poincare of Paris. Of the three classes of 
such functions (viz. functions uniform throughout, functions 
uniform only in lacunary spaces, and non-uniform functions) 
Weierstrass showed that those functions of the first class 
which can be developed according to ascending powers of x 
into converging series, can be decomposed into a product of 
an infinite number of primary factors. A primary factor of the 

species n is the product (l ) e p ( x \ P(x) being an entire 

polynomial of the nth degree. A function of the species n is 


one, all the primary factors of which are of species n. This 
classification gave rise to many interesting problems studied 
also by Poincare. 

The first of the three classes of functions of a complex 
variable embraces, among others, functions having an infinite 
number of singular points, but no singular lines, and at the 
same time no isolated singular points. These are Fuchsian 
functions, existing throughout the whole extent. Poincare 
first gave an example of such a function. 

Uniform functions of two variables, unaltered by certain 
linear substitutions, called hyperfuchsian functions, have been 
studied by E. Picard of Paris, and by Poincare. [81] 

Functions of the second class, uniform only in lacunary 
spaces, were first pointed out by Weierstrass. The Fuchsian 
and the Kleinian functions do not generally exist, except in 
the interior of a circle or of a domain otherwise bounded, 
and are therefore examples of functions of the second class. 
Poincare has shown how to generate functions of this class, and 
has studied them along the lines marked out by Weierstrass. 
Important is his proof that there is no way of generalising 
them so as to get rid of the lacunae. 

Non-uniform functions are much less developed than the 
preceding classes, even though their properties in the vicinity 
of a given point have been diligently studied, and though 
much light has been thrown on them by the use of Riemann's 
surfaces. With the view of reducing their study to that 
of uniform transcendents, Poincare proved that if y is any 


analytical non-uniform function of x, one can always find a 
variable z, such that x and y are uniform functions of z. 

Weierstrass and Darboux have each given examples of 
continuous functions having no derivatives. Formerly it had 
been generally assumed that every function had a derivative. 
Ampere was the first who attempted to prove analytically 
(1806) the existence of a derivative, but the demonstration 
is not valid. In treating of discontinuous functions, Darboux 
established rigorously the necessary and sufficient condition 
that a continuous or discontinuous function be susceptible of 
integration. He gave fresh evidence of the care that must be 
exercised in the use of series by giving an example of a series 
always convergent and continuous, such that the series formed 
by the integrals of the terms is always convergent, and yet 
does not represent the integral of the first series. [87] 

The general theory of functions of two variables has been 
investigated to some extent by Weierstrass and Poincare. 

H. A. Schwarz of Berlin (born 1845), a pupil of Weier- 
strass, has given the conform representation (Abbildung) of 
various surfaces on a circle. In transforming by aid of certain 
substitutions a polygon bounded by circular arcs into another 
also bounded by circular arcs, he was led to a remarkable 
differential equation ip(u',t) = ip(u,t), where ip(u,t) is the ex- 
pression which Cayley calls the "Schwarzian derivative," and 
which led Sylvester to the theory of reciprocants. Schwarz's 
developments on minimum surfaces, his work on hyperge- 
ometric series, his inquiries on the existence of solutions 
to important partial differential equations under prescribed 


conditions, have secured a prominent place in mathematical 

The modern theory of functions of one real variable was 
first worked out by H. Hankel, Dedekind, G. Cantor, Dini, and 
Heine, and then carried further, principally, by Weierstrass, 
Schwarz, Du Bois-Reymond, Thomae, and Darboux. Hankel 
established the principle of the condensation of singularities; 
Dedekind and Cantor gave definitions for irrational numbers; 
definite integrals were studied by Thomae, Du Bois-Reymond, 
and Darboux along the lines indicated by the definitions of 
such integrals given by Cauchy, Dirichlet, and Riemann. Dini 
wrote a text-book on functions of a real variable (1878), which 
was translated into German, with additions, by J. Liiroth and 
A. Schepp. Important works on the theory of functions are 
the Cours de M. Hermite, Tannery's Theorie des Fonctions 
d'une variable seule, A Treatise on the Theory of Functions by 
James Harkness and Frank Morley, and Theory of Functions 
of a Complex Variable by A. R. Forsyth. 


"Mathematics, the queen of the sciences, and arithmetic, 
the queen of mathematics." Such was the dictum of Gauss, 
who was destined to revolutionise the theory of numbers. 
When asked who was the greatest mathematician in Germany, 
Laplace answered, Pfaff. When the questioner said he should 
have thought Gauss was, Laplace replied, "Pfaff is by far the 
greatest mathematician in Germany; but Gauss is the greatest 


in all Europe." [83] Gauss is one of the three greatest masters 
of modern analysis, — Lagrange, Laplace, Gauss. Of these 
three contemporaries he was the youngest. While the first 
two belong to the period in mathematical history preceding 
the one now under consideration, Gauss is the one whose 
writings may truly be said to mark the beginning of our own 
epoch. In him that abundant fertility of invention, displayed 
by mathematicians of the preceding period, is combined with 
an absolute rigorousness in demonstration which is too often 
wanting in their writings, and which the ancient Greeks might 
have envied. Unlike Laplace, Gauss strove in his writings 
after perfection of form. He rivals Lagrange in elegance, 
and surpasses this great Frenchman in rigour. Wonderful 
was his richness of ideas; one thought followed another so 
quickly that he had hardly time to write down even the 
most meagre outline. At the age of twenty Gauss had 
overturned old theories and old methods in all branches of 
higher mathematics; but little pains did he take to publish his 
results, and thereby to establish his priority. He was the first 
to observe rigour in the treatment of infinite series, the first 
to fully recognise and emphasise the importance, and to make 
systematic use of determinants and of imaginaries, the first 
to arrive at the method of least squares, the first to observe 
the double periodicity of elliptic functions. He invented the 
heliotrope and, together with Weber, the bifilar magnetometer 
and the declination instrument. He reconstructed the whole 
of magnetic science. 

Carl Friedrich Gauss [47] (1777-1855), the son of a 


bricklayer, was born at Brunswick. He used to say, jokingly, 
that he could reckon before he could talk. The marvellous 
aptitude for calculation of the young boy attracted the 
attention of Bartels, afterwards professor of mathematics at 
Dorpat, who brought him under the notice of Charles William, 
Duke of Brunswick. The duke undertook to educate the boy, 
and sent him to the Collegium Carolinum. His progress in 
languages there was quite equal to that in mathematics. In 
1795 he went to Gottingen, as yet undecided whether to pursue 
philology or mathematics. Abraham Gotthelf Kastner, then 
professor of mathematics there, and now chiefly remembered 
for his Geschichte der Mathematik (1796), was not an inspiring 
teacher. At the age of nineteen Gauss discovered a method 
of inscribing in a circle a regular polygon of seventeen sides, 
and this success encouraged him to pursue mathematics. He 
worked quite independently of his teachers, and while a student 
at Gottingen made several of his greatest discoveries. Higher 
arithmetic was his favourite study. Among his small circle 
of intimate friends was Wolfgang Bolyai. After completing 
his course he returned to Brunswick. In 1798 and 1799 he 
repaired to the university at Helmstadt to consult the library, 
and there made the acquaintance of Pfaff, a mathematician 
of much power. In 1807 the Emperor of Russia offered Gauss 
a chair in the Academy at St. Petersburg, but by the advice of 
the astronomer Olbers, who desired to secure him as director 
of a proposed new observatory at Gottingen, he declined the 
offer, and accepted the place at Gottingen. Gauss had a 
marked objection to a mathematical chair, and preferred the 


post of astronomer, that he might give all his time to science. 
He spent his life in Gottingen in the midst of continuous work. 
In 1828 he went to Berlin to attend a meeting of scientists, but 
after this he never again left Gottingen, except in 1854, when 
a railroad was opened between Gottingen and Hanover. He 
had a strong will, and his character showed a curious mixture 
of self-conscious dignity and child-like simplicity. He was little 
communicative, and at times morose. 

A new epoch in the theory of numbers dates from the 
publication of his Disquisitiones Arithmeticds, Leipzig, 1801. 
The beginning of this work dates back as far as 1795. Some of 
its results had been previously given by Lagrange and Euler, 
but were reached independently by Gauss, who had gone 
deeply into the subject before he became acquainted with 
the writings of his great predecessors. The Disquisitiones 
Arithmeticde was already in print when Legendre's Theorie 
des N ombres appeared. The great law of quadratic reciprocity, 
given in the fourth section of Gauss' work, a law which involves 
the whole theory of quadratic residues, was discovered by him 
by induction before he was eighteen, and was proved by him one 
year later. Afterwards he learned that Euler had imperfectly 
enunciated that theorem, and that Legendre had attempted 
to prove it, but met with apparently insuperable difficulties. 
In the fifth section Gauss gave a second proof of this "gem" 
of higher arithmetic. In 1808 followed a third and fourth 
demonstration; in 1817, a fifth and sixth. No wonder that he 
felt a personal attachment to this theorem. Proofs were given 
also by Jacobi, Eisenstein, Liouville, Lebesgue, A. Genocchi, 


Kummer, M. A. Stern, Chr. Zeller, Kronecker, Bouniakowsky, 
E. Schering, J. Petersen, Voigt, E. Busche, and Th. Pepin. [48] 
The solution of the problem of the representation of numbers 
by binary quadratic forms is one of the great achievements of 
Gauss. He created a new algorithm by introducing the theory 
of congruences. The fourth section of the Disquisitiones 
Arithmetics, treating of congruences of the second degree, 
and the fifth section, treating of quadratic forms, were, until 
the time of Jacobi, passed over with universal neglect, but 
they have since been the starting-point of a long series of 
important researches. The seventh or last section, developing 
the theory of the division of the circle, was received from the 
start with deserved enthusiasm, and has since been repeatedly 
elaborated for students. A standard work on Kreistheilung 
was published in 1872 by Paul Bachmann, then of Breslau. 
Gauss had planned an eighth section, which was omitted to 
lessen the expense of publication. His papers on the theory of 
numbers were not all included in his great treatise. Some of 
them were published for the first time after his death in his 
collected works (1863-1871). He wrote two memoirs on the 
theory of biquadratic residues (1825 and 1831), the second of 
which contains a theorem of biquadratic reciprocity. 

Gauss was led to astronomy by the discovery of the 
planet Ceres at Palermo in 1801. His determination of 
the elements of its orbit with sufficient accuracy to enable 
Olbers to re-discover it, made the name of Gauss generally 
known. In 1809 he published the Theoria motus corporum 
coelestium, which contains a discussion of the problems 


arising in the determination of the movements of planets 
and comets from observations made on them under any 
circumstances. In it are found four formulae in spherical 
trigonometry, now usually called "Gauss' Analogies," but 
which were published somewhat earlier by Karl Brandon 
Mollweide of Leipzig (1774-1825), and earlier still by Jean 
Baptiste Joseph Delambre (1749-1822). [44] Many years of 
hard work were spent in the astronomical and magnetic 
observatory. He founded the German Magnetic Union, with 
the object of securing continuous observations at fixed times. 
He took part in geodetic observations, and in 1843 and 1846 
wrote two memoirs, Ueber Gegenstdnde derhoheren Geodesie. 
He wrote on the attraction of homogeneous ellipsoids, 1813. 
In a memoir on capillary attraction, 1833, he solves a problem 
in the calculus of variations involving the variation of a certain 
double integral, the limits of integration being also variable; 
it is the earliest example of the solution of such a problem. 
He discussed the problem of rays of light passing through a 
system of lenses. 

Among Gauss' pupils were Christian Heinrich Schumacher, 
Christian Gerling, Friedrich Nicolai, August Ferdinand 
Mobius, Georg Wilhelm Struve, Johann Frantz Encke. 

Gauss' researches on the theory of numbers were the 
starting-point for a school of writers, among the earliest of 
whom was Jacobi. The latter contributed to Crelle's Journal 
an article on cubic residues, giving theorems without proofs. 
After the publication of Gauss' paper on biquadratic residues, 
giving the law of biquadratic reciprocity, and his treatment 


of complex numbers, Jacobi found a similar law for cubic 
residues. By the theory of elliptical functions, he was led 
to beautiful theorems on the representation of numbers by 
2, 4, 6, and 8 squares. Next come the researches of Dirichlet, 
the expounder of Gauss, and a contributor of rich results of 
his own. 

Peter Gustav Lejeune Dirichlet [88] (1805-1859) was 
born in Diiren, attended the gymnasium in Bonn, and then 
the Jesuit gymnasium in Cologne. In 1822 he was attracted 
to Paris by the names of Laplace, Legendre, Fourier, Poisson, 
Cauchy. The facilities for a mathematical education there 
were far better than in Germany, where Gauss was the 
only great figure. He read in Paris Gauss' Disquisitiones 
ArithmeticcB, a work which he never ceased to admire and 
study. Much in it was simplified by Dirichlet, and thereby 
placed within easier reach of mathematicians. His first memoir 
on the impossibility of certain indeterminate equations of the 
fifth degree was presented to the French Academy in 1825. 
He showed that Fermat's equation, x n + y n = z n , cannot 
exist when n = 5. Some parts of the analysis are, however, 
Legendre's. Euler and Lagrange had proved this when 
n is 3 and 4, and Lame proved it when n = 7. Dirichlet's 
acquaintance with Fourier led him to investigate Fourier's 
series. He became docent in Breslau in 1827. In 1828 he 
accepted a position in Berlin, and finally succeeded Gauss at 
Gottingen in 1855. The general principles on which depends 
the average number of classes of binary quadratic forms of 
positive and negative determinant (a subject first investigated 


by Gauss) were given by Dirichlet in a memoir, Ueber 
die Bestimmung der mittleren Werthe in der Zahlentheorie, 
1849. More recently F. Mertens of Graz has determined the 
asymptotic values of several numerical functions. Dirichlet 
gave some attention to prime numbers. Gauss and Legendre 
had given expressions denoting approximately the asymptotic 
value of the number of primes inferior to a given limit, but it 
remained for Riemann in his memoir, Ueber die Anzahl der 
Primzahlen unter einer gegebenen Grosse, 1859, to give an 
investigation of the asymptotic frequency of primes which is 
rigorous. Approaching the problem from a different direction, 
Patnutij Tchebycheff, formerly professor in the University 
of St. Petersburg (born 1821), established, in a celebrated 
memoir, Sur les Nombres Premiers, 1850, the existence of 
limits within which the sum of the logarithms of the primes P, 
inferior to a given number x, must be comprised. [89] This 
paper depends on very elementary considerations, and, in that 
respect, contrasts strongly with Riemann's, which involves 
abstruse theorems of the integral calculus. Poincare's papers, 
Sylvester's contraction of Tchebycheff's limits, with reference 
to the distribution of primes, and researches of J. Hadamard 
(awarded the Grand prix of 1892), are among the latest 
researches in this line. The enumeration of prime numbers has 
been undertaken at different times by various mathematicians. 
In 1877 the British Association began the preparation of 
factor-tables, under the direction of J. W. L. Glaisher. The 
printing, by the Association, of tables for the sixth million 
marked the completion of tables, to the preparation of which 


Germany, France, and England contributed, and which enable 
us to resolve into prime factors every composite number less 
than 9, 000, 000. 

Miscellaneous contributions to the theory of numbers were 
made by Cauchy. He showed, for instance, how to find all the 
infinite solutions of a homogeneous indeterminate equation 
of the second degree in three variables when one solution is 
given. He established the theorem that if two congruences, 
which have the same modulus, admit of a common solution, 
the modulus is a divisor of their resultant. Joseph Liouville 
(1809-1882), professor at the College de France, investigated 
mainly questions on the theory of quadratic forms of two, 
and of a greater number of variables. Profound researches 
were instituted by Ferdinand Gotthold Eisenstein (1823- 
1852), of Berlin. Ternary quadratic forms had been studied 
somewhat by Gauss, but the extension from two to three 
indeterminates was the work of Eisenstein who, in his memoir, 
Neue Theoreme der hoheren Arithmetik, defined the ordinal 
and generic characters of ternary quadratic forms of uneven 
determinant; and, in case of definite forms, assigned the weight 
of any order or genus. But he did not publish demonstrations 
of his results. In inspecting the theory of binary cubic forms, 
he was led to the discovery of the first covariant ever considered 
in analysis. He showed that the series of theorems, relating to 
the presentation of numbers by sums of squares, ceases when 
the number of squares surpasses eight. Many of the proofs 
omitted by Eisenstein were supplied by Henry Smith, who 
was one of the few Englishmen who devoted themselves to the 


study of higher arithmetic. 

Henry John Stephen Smith [90] (1826-1883) was born 
in London, and educated at Rugby and at Balliol College, 
Oxford. Before 1847 he travelled much in Europe for his 
health, and at one time attended lectures of Arago in Paris, 
but after that year he was never absent from Oxford for a single 
term. In 1861 he was elected Savilian professor of geometry. 
His first paper on the theory of numbers appeared in 1855. 
The results of ten years' study of everything published on 
the theory of numbers are contained in his Reports which 
appeared in the British Association volumes from 1859 to 1865. 
These reports are a model of clear and precise exposition and 
perfection of form. They contain much original matter, but 
the chief results of his own discoveries were printed in the 
Philosophical Transactions for 1861 and 1867. They treat of 
linear indeterminate equations and congruences, and of the 
orders and genera of ternary quadratic forms. He established 
the principles on which the extension to the general case of n 
indeterminates of quadratic forms depends. He contributed 
also two memoirs to the Proceedings of the Royal Society of 
1864 and 1868, in the second of which he remarks that the 
theorems of Jacobi, Eisenstein, and Liouville, relating to the 
representation of numbers by 4, 6, 8 squares, and other simple 
quadratic forms are deducible by a uniform method from the 
principles indicated in his paper. Theorems relating to the case 
of 5 squares were given by Eisenstein, but Smith completed the 
enunciation of them, and added the corresponding theorems 
for 7 squares. The solution of the cases of 2, 4, 6 squares 


may be obtained by elliptic functions, but when the number 
of squares is odd, it involves processes peculiar to the theory 
of numbers. This class of theorems is limited to 8 squares, 
and Smith completed the group. In ignorance of Smith's 
investigations, the French Academy offered a prize for the 
demonstration and completion of Eisenstein's theorems for 
5 squares. This Smith had accomplished fifteen years earlier. 
He sent in a dissertation in 1882, and next year, a month 
after his death, the prize was awarded to him, another prize 
being also awarded to H. Minkowsky of Bonn. The theory of 
numbers led Smith to the study of elliptic functions. He wrote 
also on modern geometry. His successor at Oxford was J.J. 

Ernst Eduard Kummer (1810-1893), professor in the 
University of Berlin, is closely identified with the theory 
of numbers. Dirichlet's work on complex numbers of the 
form a + ib, introduced by Gauss, was extended by him, by 
Eisenstein, and Dedekind. Instead of the equation x 4 — 1 = 0, 
the roots of which yield Gauss' units, Eisenstein used the 
equation x 3 — 1 = and complex numbers a + bp(p being a cube 
root of unity), the theory of which resembles that of Gauss' 
numbers. Kummer passed to the general case x n — 1 = and 

got complex numbers of the form a = a\Ai + a2A2 + a^A^-\ , 

where aj are whole real numbers, and A$ roots of the above 
equation. [59] Euclid's theory of the greatest common divisor 
is not applicable to such complex numbers, and their prime 
factors cannot be defined in the same way as prime factors 
of common integers are defined. In the effort to overcome 


this difficulty, Kummer was led to introduce the conception 
of "ideal numbers." These ideal numbers have been applied 
by G. Zolotareff of St. Petersburg to the solution of a 
problem of the integral calculus, left unfinished by Abel 
(Liouville's Journal, Second Series, 1864, Vol. IX.). Julius 
Wilhelm Richard Dedekind of Braunschweig (born 1831) 
has given in the second edition of Dirichlet's Vorlesungen ilber 
Zahlentheorie a new theory of complex numbers, in which 
he to some extent deviates from the course of Kummer, and 
avoids the use of ideal numbers. Dedekind has taken the 
roots of any irreducible equation with integral coefficients as 
the units for his complex numbers. Attracted by Kummer's 
investigations, his pupil, Leopold Kronecker (1823-1891) 
made researches which he applied to algebraic equations. 

On the other hand, efforts have been made to utilise in the 
theory of numbers the results of the modern higher algebra. 
Following up researches of Hermite, Paul Bachmann of 
Munster investigated the arithmetical formula which gives 
the automorphics of a ternary quadratic form. [89] The 
problem of the equivalence of two positive or definite ternary 
quadratic forms was solved by L. Seeber; and that of the 
arithmetical automorphics of such forms, by Eisenstein. The 
more difficult problem of the equivalence for indefinite ternary 
forms has been investigated by Edward Selling of Wiirzburg. 
On quadratic forms of four or more indeterminates little 
has yet been done. Hermite showed that the number of 
non-equivalent classes of quadratic forms having integral 
coefficients and a given discriminant is finite, while Zolotareff 


and A. N. Korkine, both of St. Petersburg, investigated the 
minima of positive quadratic forms. In connection with binary 
quadratic forms, Smith established the theorem that if the 
joint invariant of two properly primitive forms vanishes, the 
determinant of either of them is represented primitively by 
the duplicate of the other. 

The interchange of theorems between arithmetic and alge- 
bra is displayed in the recent researches of J. W. L. Glaisher 
of Trinity College (born 1848) and Sylvester. Sylvester gave a 
Constructive Theory of Partitions, which received additions 
from his pupils, F. Franklin and G. S. Ely. 

The conception of "number" has been much extended in our 
time. With the Greeks it included only the ordinary positive 
whole numbers; Diophantus added rational fractions to the 
domain of numbers. Later negative numbers and imaginaries 
came gradually to be recognised. Descartes fully grasped the 
notion of the negative; Gauss, that of the imaginary. With 
Euclid, a ratio, whether rational or irrational, was not a 
number. The recognition of ratios and irrationals as numbers 
took place in the sixteenth century, and found expression 
with Newton. By the ratio method, the continuity of the real 
number system has been based on the continuity of space, but 
in recent time three theories of irrationals have been advanced 
by Weierstrass, J. W. R. Dedekind, G. Cantor, and Heine, 
which prove the continuity of numbers without borrowing it 
from space. They are based on the definition of numbers by 
regular sequences, the use of series and limits, and some new 
mathematical conceptions. 



Notwithstanding the beautiful developments of celestial 
mechanics reached by Laplace at the close of the eighteenth 
century, there was made a discovery on the first day of the 
present century which presented a problem seemingly beyond 
the power of that analysis. We refer to the discovery of 
Ceres by Piazzi in Italy, which became known in Germany 
just after the philosopher Hegel had published a dissertation 
proving a priori that such a discovery could not be made. 
From the positions of the planet observed by Piazzi its orbit 
could not be satisfactorily calculated by the old methods, and 
it remained for the genius of Gauss to devise a method of 
calculating elliptic orbits which was free from the assumption 
of a small eccentricity and inclination. Gauss' method 
was developed further in his Theoria Motus. The new 
planet was re-discovered with aid of Gauss' data by Olbers, 
an astronomer who promoted science not only by his own 
astronomical studies, but also by discerning and directing 
towards astronomical pursuits the genius of Bessel. 

Friedrich Wilhelm Bessel [91] (1784-1846) was a native 
of Minden in Westphalia. Fondness for figures, and a distaste 
for Latin grammar led him to the choice of a mercantile 
career. In his fifteenth year he became an apprenticed clerk 
in Bremen, and for nearly seven years he devoted his days to 
mastering the details of his business, and part of his nights 
to study. Hoping some day to become a supercargo on 
trading expeditions, he became interested in observations at 


sea. With a sextant constructed by him and an ordinary clock 
he determined the latitude of Bremen. His success in this 
inspired him for astronomical study. One work after another 
was mastered by him, unaided, during the hours snatched 
from sleep. From old observations he calculated the orbit 
of Halley's comet. Bessel introduced himself to Olbers, and 
submitted to him the calculation, which Olbers immediately 
sent for publication. Encouraged by Olbers, Bessel turned 
his back to the prospect of affluence, chose poverty and the 
stars, and became assistant in J. H. Schroter's observatory 
at Lilienthal. Four years later he was chosen to superintend 
the construction of the new observatory at Konigsberg. [92] 
In the absence of an adequate mathematical teaching force, 
Bessel was obliged to lecture on mathematics to prepare 
students for astronomy. He was relieved of this work in 1825 
by the arrival of Jacobi. We shall not recount the labours by 
which Bessel earned the title of founder of modern practical 
astronomy and geodesy. As an observer he towered far above 
Gauss, but as a mathematician he reverently bowed before 
the genius of his great contemporary. Of Bessel's papers, the 
one of greatest mathematical interest is an " Untersuchung 
des Theils der planetarischen Storungen, welcher aus der 
Bewegung der Sonne ensteht" (1824), in which he introduces 
a class of transcendental functions, J n {x), much used in 
applied mathematics, and known as "Bessel's functions." He 
gave their principal properties, and constructed tables for 
their evaluation. Recently it has been observed that Bessel's 
functions appear much earlier in mathematical literature. [98] 


Such functions of the zero order occur in papers of Daniel 
Bernoulli (1732) and Euler on vibration of heavy strings 
suspended from one end. All of Bessel's functions of the first 
kind and of integral orders occur in a paper by Euler (1764) on 
the vibration of a stretched elastic membrane. In 1878 Lord 
Rayleigh proved that Bessel's functions are merely particular 
cases of Laplace's functions. J. W. L. Glaisher illustrates by 
Bessel's functions his assertion that mathematical branches 
growing out of physical inquiries as a rule "lack the easy 
flow or homogeneity of form which is characteristic of a 
mathematical theory properly so called." These functions 
have been studied by C. Th. Anger of Danzig, O. Schlomilch of 
Dresden, R. Lipschitz of Bonn (born 1832), Carl Neumann of 
Leipzig (born 1832), Eugen Lommel of Leipzig, I. Todhunter 
of St. John's College, Cambridge. 

Prominent among the successors of Laplace are the follow- 
ing: Simeon Denis Poisson (1781-1840), who wrote in 1808 
a classic Memoire sur les inegalites seculaires des moyens 
mouvements des planetes. Giovanni Antonio Amadeo Plana 
(1781-1864) of Turin, a nephew of Lagrange, who published 
in 1811 a Memoria sulla teoria dell' attrazione degli sferoidi 
ellitici, and contributed to the theory of the moon. Peter 
Andreas Hansen (1795-1874) of Gotha, at one time a 
clockmaker in Tondern, then Schumacher's assistant at Al- 
tona, and finally director of the observatory at Gotha, wrote 
on various astronomical subjects, but mainly on the lunar 
theory, which he elaborated in his work Fundamenta nova 
investigationes orbitce verce quam Luna perlustrat (1838), and 


in subsequent investigations embracing extensive lunar ta- 
bles. George Biddel Airy (1801-1892), royal astronomer at 
Greenwich, published in 1826 his Mathematical Tracts on the 
Lunar and Planetary Theories. These researches have since 
been greatly extended by him. August Ferdinand Mobius 
(1790-1868) of Leipzig wrote, in 1842, Elemente der Mechanik 
des Himmels. Urbain Jean Joseph Le Verrier (1811-1877) 
of Paris wrote the Recherches Astronomiques, constituting 
in part a new elaboration of celestial mechanics, and is fa- 
mous for his theoretical discovery of Neptune. John Couch 
Adams (1819-1892) of Cambridge divided with Le Verrier 
the honour of the mathematical discovery of Neptune, and 
pointed out in 1853 that Laplace's explanation of the secular 
acceleration of the moon's mean motion accounted for only 
half the observed acceleration. Charles Eugene Delaunay 
(born 1816, and drowned off Cherbourg in 1872), professor 
of mechanics at the Sorbonne in Paris, explained most of 
the remaining acceleration of the moon, unaccounted for by 
Laplace's theory as corrected by Adams, by tracing the effect 
of tidal friction, a theory previously suggested independently 
by Kant, Robert Mayer, and William Ferrel of Kentucky. 
George Howard Darwin of Cambridge (born 1845) made 
some very remarkable investigations in 1879 on tidal friction, 
which trace with great certainty the history of the moon from 
its origin. He has since studied also the effects of tidal friction 
upon other bodies in the solar system. Criticisms on some 
parts of his researches have been made by James Nolan of 
Victoria. Simon Newcomb (born 1835), superintendent 


of the Nautical Almanac at Washington, and professor of 
mathematics at the Johns Hopkins University, investigated 
the errors in Hansen's tables of the moon. For the last twelve 
years the main work of the U. S. Nautical Almanac office has 
been to collect and discuss data for new tables of the planets 
which will supplant the tables of Le Verrier . G. W. Hill of that 
office has contributed an elegant paper on certain possible 
abbreviations in the computation of the long-period of the 
moon's motion due to the direct action of the planets, and 
has made the most elaborate determination yet undertaken of 
the inequalities of the moon's motion due to the figure of the 
earth. He has also computed certain lunar inequalities due to 
the action of Jupiter. 

The mathematical discussion of Saturn's rings was taken up 
first by Laplace, who demonstrated that a homogeneous solid 
ring could not be in equilibrium, and in 1851 by B. Peirce, who 
proved their non-solidity by showing that even an irregular 
solid ring could not be in equilibrium about Saturn. The 
mechanism of these rings was investigated by James Clerk 
Maxwell in an essay to which the Adams prize was awarded. He 
concluded that they consisted of an aggregate of unconnected 

The problem of three bodies has been treated in various 
ways since the time of Lagrange, but no decided advance 
towards a more complete algebraic solution has been made, 
and the problem stands substantially where it was left by him. 
He had made a reduction in the differential equations to the 
seventh order. This was elegantly accomplished in a different 


way by Jacobi in 1843. R. Radau (Comptes Rendus, LXVIL, 
1868, p. 841) and Allegret [Journal de Mathematiques, 1875, 
p. 277) showed that the reduction can be performed on the 
equations in their original form. Noteworthy transformations 
and discussions of the problem have been given by J. L. F. 
Bertrand, by Emile Bour (1831-1866) of the Polytechnic 
School in Paris, by Mathieu, Hesse, J. A. Serret. H. Bruns of 
Leipzig has shown that no advance in the problem of three or 
of n bodies may be expected by algebraic integrals, and that 
we must look to the modern theory of functions for a complete 
solution {Acta Math., XL, p. 43). [93] 

Among valuable text-books on mathematical astronomy 
rank the following works: Manual of Spherical and Practical 
Astronomy by Chauvenet (1863), Practical and Spherical 
Astronomy by Robert Main of Cambridge, Theoretical As- 
tronomy by James C. Watson of Ann Arbor (1868), Traite 
elementaire de Mecanique Celeste of H. Resal of the Poly- 
technic School in Paris, Cours d'Astronomie de I'Ecole Poly- 
technique by Faye, Traite de Mecanique Celeste by Tisserand, 
Lehrbuch der Bahnbestimmung by T. Oppolzer, Mathematis- 
che Theorien der Planetenbewegung by 0. Dziobek, translated 
into English by M. W. Harrington and W. J. Hussey. 

During the present century we have come to recognise the 
advantages frequently arising from a geometrical treatment 
of mechanical problems. To Poinsot, Chasles, and Mobius we 
owe the most important developments made in geometrical 
mechanics. Louis Poinsot (1777-1859), a graduate of the 
Polytechnic School in Paris, and for many years member of 


the superior council of public instruction, published in 1804 
his Elements de Statique. This work is remarkable not only 
as being the earliest introduction to synthetic mechanics, but 
also as containing for the first time the idea of couples, which 
was applied by Poinsot in a publication of 1834 to the theory 
of rotation. A clear conception of the nature of rotary motion 
was conveyed by Poinsot 's elegant geometrical representation 
by means of an ellipsoid rolling on a certain fixed plane. This 
construction was extended by Sylvester so as to measure the 
rate of rotation of the ellipsoid on the plane. 

A particular class of dynamical problems has recently been 
treated geometrically by Sir Robert Stawell Ball, formerly 
astronomer royal of Ireland, now Lowndean Professor of 
Astronomy and Geometry at Cambridge. His method is 
given in a work entitled Theory of Screws, Dublin, 1876, 
and in subsequent articles. Modern geometry is here drawn 
upon, as was done also by Clifford in the related subject of 
Biquaternions. Arthur Buchheim of Manchester (1859-1888), 
showed that Grassmann's Ausdehnungslehre supplies all the 
necessary materials for a simple calculus of screws in elliptic 
space. Horace Lamb applied the theory of screws to the 
question of the steady motion of any solid in a fluid. 

Advances in theoretical mechanics, bearing on the integra- 
tion and the alteration in form of dynamical equations, were 
made since Lagrange by Poisson, William Rowan Hamilton, 
Jacobi, Madame Kowalevski, and others. Lagrange had estab- 
lished the "Lagrangian form" of the equations of motion. He 
had given a theory of the variation of the arbitrary constants 


which, however, turned out to be less fruitful in results than a 
theory advanced by Poisson. [99] Poisson's theory of the vari- 
ation of the arbitrary constants and the method of integration 
thereby afforded marked the first onward step since Lagrange. 
Then came the researches of Sir William Rowan Hamilton. 
His discovery that the integration of the dynamic differential 
equations is connected with the integration of a certain partial 
differential equation of the first order and second degree, grew 
out of an attempt to deduce, by the undulatory theory, results 
in geometrical optics previously based on the conceptions of 
the emission theory. The Philosophical Transactions of 1833 
and 1834 contain Hamilton's papers, in which appear the first 
applications to mechanics of the principle of varying action 
and the characteristic function, established by him some years 
previously. The object which Hamilton proposed to himself 
is indicated by the title of his first paper, viz. the discovery 
of a function by means of which all integral equations can be 
actually represented. The new form obtained by him for the 
equation of motion is a result of no less importance than that 
which was the professed object of the memoir. Hamilton's 
method of integration was freed by Jacobi of an unnecessary 
complication, and was then applied by him to the determina- 
tion of a geodetic line on the general ellipsoid. With aid of 
elliptic co-ordinates Jacobi integrated the partial differential 
equation and expressed the equation of the geodetic in form 
of a relation between two Abelian integrals. Jacobi applied 
to differential equations of dynamics the theory of the ulti- 
mate multiplier. The differential equations of dynamics are 


only one of the classes of differential equations considered by 
Jacobi. Dynamic investigations along the lines of Lagrange, 
Hamilton, and Jacobi were made by Liouville, A. Desboves, 
Serret, J. C. F. Sturm, Ostrogradsky, J. Bertrand, Donkin, 
Brioschi, leading up to the development of the theory of a 
system of canonical integrals. 

An important addition to the theory of the motion of a solid 
body about a fixed point was made by Madame Sophie de 
Kowalevski [96] (1853-1891), who discovered a new case in 
which the differential equations of motion can be integrated. 
By the use of theta- functions of two independent variables she 
furnished a remarkable example of how the modern theory of 
functions may become useful in mechanical problems. She 
was a native of Moscow, studied under Weierstrass, obtained 
the doctor's degree at Gottingen, and from 1884 until her 
death was professor of higher mathematics at the University 
of Stockholm. The research above mentioned received the 
Bordin prize of the French Academy in 1888, which was 
doubled on account of the exceptional merit of the paper. 

There are in vogue three forms for the expression of the 
kinetic energy of a dynamical system: the Lagrangian, the 
Hamiltonian, and a modified form of Lagrange's equations 
in which certain velocities are omitted. The kinetic energy 
is expressed in the first form as a homogeneous quadratic 
function of the velocities, which are the time-variations of 
the co-ordinates of the system; in the second form, as 
a homogeneous quadratic function of the momenta of the 
system; the third form, elaborated recently by Edward 


John Routh of Cambridge, in connection with his theory 
of "ignoration of co-ordinates," and by A. B. Basset, is 
of importance in hydrodynamical problems relating to the 
motion of perforated solids in a liquid, and in other branches 
of physics. 

In recent time great practical importance has come to be 
attached to the principle of mechanical similitude. By it one 
can determine from the performance of a model the action 
of the machine constructed on a larger scale. The principle 
was first enunciated by Newton (Principia, Bk. II., Sec. VIII., 
Prop. 32), and was derived by Bertrand from the principle of 
virtual velocities. A corollary to it, applied in ship-building, 
goes by the name of William Froude's law, but was enunciated 
also by Reech. 

The present problems of dynamics differ materially from 
those of the last century. The explanation of the orbital and 
axial motions of the heavenly bodies by the law of universal 
gravitation was the great problem solved by Clairaut, Euler, 
D'Alembert, Lagrange, and Laplace. It did not involve 
the consideration of frictional resistances. In the present 
time the aid of dynamics has been invoked by the physical 
sciences. The problems there arising are often complicated 
by the presence of friction. Unlike astronomical problems 
of a century ago, they refer to phenomena of matter and 
motion that are usually concealed from direct observation. 
The great pioneer in such problems is Lord Kelvin. While yet 
an undergraduate at Cambridge, during holidays spent at the 
seaside, he entered upon researches of this kind by working 


out the theory of spinning tops, which previously had been 
only partially explained by Jellet in his Treatise on the Theory 
of Friction (1872), and by Archibald Smith. 

Among standard works on mechanics are Jacobi's Vor- 
lesungen ilber Dynamik, edited by Clebsch, 1866; Kirchhoff' s 
Vorlesungen iiber mathematische Physik, 1876; Benjamin 
Peirce's Analytic Mechanics, 1855; Somoff' s Theoretische 
Mechanik, 1879; Tait and Steele's Dynamics of a Particle, 
1856; Minchin's Treatise on Statics; Routh's Dynamics of 
a System of Rigid Bodies; Sturm's Cours de Mecanique de 
I'Ecole Polytechnique. 

The equations which constitute the foundation of the theory 
of fluid motion were fully laid down at the time of Lagrange, 
but the solutions actually worked out were few and mainly 
of the irrotational type. A powerful method of attacking 
problems in fluid motion is that of images, introduced in 1843 
by George Gabriel Stokes of Pembroke College, Cambridge. It 
received little attention until Sir William Thomson's discovery 
of electrical images, whereupon the theory was extended by 
Stokes, Hicks, and Lewis. In 1849, Thomson gave the 
maximum and minimum theorem peculiar to hydrodynamics, 
which was afterwards extended to dynamical problems in 

A new epoch in the progress of hydrodynamics was created, 
in 1856, by Helmholtz, who worked out remarkable properties 
of rotational motion in a homogeneous, incompressible fluid, 
devoid of viscosity. He showed that the vortex filaments in 
such a medium may possess any number of knottings and 


twistings, but are either endless or the ends are in the free 
surface of the medium; they are indivisible. These results 
suggested to Sir William Thomson the possibility of founding 
on them a new form of the atomic theory, according to 
which every atom is a vortex ring in a non-frictional ether, 
and as such must be absolutely permanent in substance and 
duration. The vortex-atom theory is discussed by J. J. 
Thomson of Cambridge (born 1856) in his classical treatise 
on the Motion of Vortex Rings, to which the Adams Prize 
was awarded in 1882. Papers on vortex motion have been 
published also by Horace Lamb, Thomas Craig, Henry A. 
Rowland, and Charles Chree. 

The subject of jets was investigated by Helmholtz, Kirch- 
hoff, Plateau, and Rayleigh; the motion of fluids in a fluid 
by Stokes, Sir W. Thomson, Kopcke, Greenhill, and Lamb; 
the theory of viscous fluids by Navier, Poisson, Saint- Venant, 
Stokes, O. E. Meyer, Stefano, Maxwell, Lipschitz, Craig, 
Helmholtz, and A. B. Basset. Viscous fluids present great 
difficulties, because the equations of motion have not the 
same degree of certainty as in perfect fluids, on account of a 
deficient theory of friction, and of the difficulty of connecting 
oblique pressures on a small area with the differentials of the 

Waves in liquids have been a favourite subject with English 
mathematicians. The early inquiries of Poisson and Cauchy 
were directed to the investigation of waves produced by 
disturbing causes acting arbitrarily on a small portion of the 
fluid. The velocity of the long wave was given approximately 


by Lagrange in 1786 in case of a channel of rectangular 
cross-section, by Green in 1839 for a channel of triangular 
section, and by P. Kelland for a channel of any uniform 
section. Sir George B. Airy, in his treatise on Tides and 
Waves, discarded mere approximations, and gave the exact 
equation on which the theory of the long wave in a channel 
of uniform rectangular section depends. But he gave no 
general solutions. J. McCowan of University College at 
Dundee discusses this topic more fully, and arrives at exact 
and complete solutions for certain cases. The most important 
application of the theory of the long wave is to the explanation 
of tidal phenomena in rivers and estuaries. 

The mathematical treatment of solitary waves was first 
taken up by S. Earnshaw in 1845, then by Stokes; but the 
first sound approximate theory was given by J. Boussinesq in 
1871, who obtained an equation for their form, and a value for 
the velocity in agreement with experiment. Other methods 
of approximation were given by Rayleigh and J. McCowan. 
In connection with deep-water waves, Osborne Reynolds gave 
in 1877 the dynamical explanation for the fact that a group 
of such waves advances with only half the rapidity of the 
individual waves. 

The solution of the problem of the general motion of an 
ellipsoid in a fluid is due to the successive labours of Green 
(1833), Clebsch (1856), and Bjerknes (1873). The free motion 
of a solid in a liquid has been investigated by W. Thomson, 
Kirchhoff, and Horace Lamb. By these labours, the motion of 
a single solid in a fluid has come to be pretty well understood, 


but the case of two solids in a fluid is not developed so fully. 
The problem has been attacked by W. M. Hicks. 

The determination of the period of oscillation of a rotating 
liquid spheroid has important bearings on the question of the 
origin of the moon. G. H. Darwin's investigations thereon, 
viewed in the light of Riemann's and Poincare's researches, 
seem to disprove Laplace's hypothesis that the moon separated 
from the earth as a ring, because the angular velocity was too 
great for stability; Darwin finds no instability. 

The explanation of the contracted vein has been a point of 
much controversy, but has been put in a much better light by 
the application of the principle of momentum, originated by 
Froude and Rayleigh. Rayleigh considered also the reflection 
of waves, not at the surface of separation of two uniform 
media, where the transition is abrupt, but at the confines of 
two media between which the transition is gradual. 

The first serious study of the circulation of winds on the 
earth's surface was instituted at the beginning of the second 
quarter of this century by H. W. Dove, William C. Redfield, and 
James P. Espy, followed by researches of W. Reid, Piddington, 
and Elias Loomis. But the deepest insight into the wonderful 
correlations that exist among the varied motions of the 
atmosphere was obtained by William Ferrel (1817-1891). 
He was born in Fulton County, Pa., and brought up on a farm. 
Though in unfavourable surroundings, a burning thirst for 
knowledge spurred the boy to the mastery of one branch after 
another. He attended Marshall College, Pa., and graduated in 
1844 from Bethany College. While teaching school he became 


interested in meteorology and in the subject of tides. In 1856 
he wrote an article on "the winds and currents of the ocean." 
The following year he became connected with the Nautical 
Almanac. A mathematical paper followed in 1858 on "the 
motion of fluids and solids relative to the earth's surface." 
The subject was extended afterwards so as to embrace the 
mathematical theory of cyclones, tornadoes, water-spouts, 
etc. In 1885 appeared his Recent Advances in Meteorology. In 
the opinion of a leading European meteorologist (Julius Hann 
of Vienna), Ferrel has "contributed more to the advance of 
the physics of the atmosphere than any other living physicist 
or meteorologist." 

Ferrel teaches that the air flows in great spirals toward the 
poles, both in the upper strata of the atmosphere and on 
the earth's surface beyond the 30th degree of latitude; while 
the return current blows at nearly right angles to the above 
spirals, in the middle strata as well as on the earth's surface, 
in a zone comprised between the parallels 30° N. and 30° S. 
The idea of three superposed currents blowing spirals was 
first advanced by James Thomson, but was published in very 
meagre abstract. 

Ferrel's views have given a strong impulse to theoretical re- 
search in America, Austria, and Germany. Several objections 
raised against his argument have been abandoned, or have 
been answered by W. M. Davis of Harvard. The mathematical 
analysis of F. Waldo of Washington, and of others, has further 
confirmed the accuracy of the theory. The transport of 
Krakatoa dust and observations made on clouds point toward 


the existence of an upper east current on the equator, and 
Pernter has mathematically deduced from Ferrel's theory the 
existence of such a current. 

Another theory of the general circulation of the atmosphere 
was propounded by Werner Siemens of Berlin, in which an 
attempt is made to apply thermodynamics to aerial currents. 
Important new points of view have been introduced recently 
by Helmholtz, who concludes that when two air currents blow 
one above the other in different directions, a system of air 
waves must arise in the same way as waves are formed on the 
sea. He and A. Oberbeck showed that when the waves on the 
sea attain lengths of from 16 to 33 feet, the air waves must 
attain lengths of from 10 to 20 miles, and proportional depths. 
Superposed strata would thus mix more thoroughly, and their 
energy would be partly dissipated. From hydrodynamical 
equations of rotation Helmholtz established the reason why 
the observed velocity from equatorial regions is much less 
in a latitude of, say, 20° or 30°, than it would be were the 
movements unchecked. 

About 1860 acoustics began to be studied with renewed 
zeal. The mathematical theory of pipes and vibrating strings 
had been elaborated in the eighteenth century by Daniel 
Bernoulli, D'Alembert, Euler, and Lagrange. In the first part 
of the present century Laplace corrected Newton's theory on 
the velocity of sound in gases, Poisson gave a mathematical 
discussion of torsional vibrations; Poisson, Sophie Germain, 
and Wheatstone studied Chladni's figures; Thomas Young 
and the brothers Weber developed the wave-theory of sound. 


Sir J. F. W. Herschel wrote on the mathematical theory 
of sound for the Encyclopaedia Metropolitana, 1845. Epoch- 
making were Helmholtz's experimental and mathematical 
researches. In his hands and Rayleigh's, Fourier's series 
received due attention. Helmholtz gave the mathematical 
theory of beats, difference tones, and summation tones. Lord 
Rayleigh (John William Strutt) of Cambridge (born 1842) 
made extensive mathematical researches in acoustics as a part 
of the theory of vibration in general. Particular mention may 
be made of his discussion of the disturbance produced by a 
spherical obstacle on the waves of sound, and of phenomena, 
such as sensitive flames, connected with the instability of 
jets of fluid. In 1877 and 1878 he published in two volumes 
a treatise on The Theory of Sound. Other mathematical 
researches on this subject have been made in England by 
Donkin and Stokes. 

The theory of elasticity [42] belongs to this century. Before 
1800 no attempt had been made to form general equations 
for the motion or equilibrium of an elastic solid. Particular 
problems had been solved by special hypotheses. Thus, 
James Bernoulli considered elastic laminae; Daniel Bernoulli 
and Euler investigated vibrating rods; Lagrange and Euler, 
the equilibrium of springs and columns. The earliest in- 
vestigations of this century, by Thomas Young ("Young's 
modulus of elasticity" ) in England, J. Binet in France, and 
G. A. A. Plana in Italy, were chiefly occupied in extend- 
ing and correcting the earlier labours. Between 1830 and 
1840 the broad outline of the modern theory of elasticity 


was established. This was accomplished almost exclusively 
by French writers, — Louis-Marie-Henri Navier (1785-1836), 
Poisson, Cauchy, Mademoiselle Sophie Germain (1776-1831), 
Felix Savart (1791-1841). 

Simeon Denis Poisson [94] (1781-1840) was born at 
Pithiviers. The boy was put out to a nurse, and he used to tell 
that when his father (a common soldier) came to see him one 
day, the nurse had gone out and left him suspended by a thin 
cord to a nail in the wall in order to protect him from perishing 
under the teeth of the carnivorous and unclean animals that 
roamed on the floor. Poisson used to add that his gymnastic 
efforts when thus suspended caused him to swing back and 
forth, and thus to gain an early familiarity with the pendulum, 
the study of which occupied him much in his maturer life. 
His father destined him for the medical profession, but so 
repugnant was this to him that he was permitted to enter 
the Polytechnic School at the age of seventeen. His talents 
excited the interest of Lagrange and Laplace. At eighteen 
he wrote a memoir on finite differences which was printed on 
the recommendation of Legendre. He soon became a lecturer 
at the school, and continued through life to hold various 
government scientific posts and professorships. He prepared 
some 400 publications, mainly on applied mathematics. His 
Traite de Mecanique, 2 vols., 1811 and 1833, was long a 
standard work. He wrote on the mathematical theory of heat, 
capillary action, probability of judgment, the mathematical 
theory of electricity and magnetism, physical astronomy, the 
attraction of ellipsoids, definite integrals, series, and the 


theory of elasticity. He was considered one of the leading 
analysts of his time. 

His work on elasticity is hardly excelled by that of Cauchy, 
and second only to that of Saint- Venant. There is hardly 
a problem in elasticity to which he has not contributed, 
while many of his inquiries were new. The equilibrium and 
motion of a circular plate was first successfully treated by 
him. Instead of the definite integrals of earlier writers, he used 
preferably finite summations. Poisson's contour conditions 
for elastic plates were objected to by Gustav Kirchhoff of 
Berlin, who established new conditions. But Thomson and 
Tait in their Treatise on Natural Philosophy have explained 
the discrepancy between Poisson's and Kirchhoff 's boundary 
conditions, and established a reconciliation between them. 

Important contributions to the theory of elasticity were 
made by Cauchy. To him we owe the origin of the theory of 
stress, and the transition from the consideration of the force 
upon a molecule exerted by its neighbours to the consideration 
of the stress upon a small plane at a point. He anticipated 
Green and Stokes in giving the equations of isotropic elasticity 
with two constants. The theory of elasticity was presented by 
Gabrio Piola of Italy according to the principles of Lagrange's 
Mecanique Analytique, but the superiority of this method over 
that of Poisson and Cauchy is far from evident. The influence of 
temperature on stress was first investigated experimentally by 
Wilhelm Weber of Gottingen, and afterwards mathematically 
by Duhamel, who, assuming Poisson's theory of elasticity, 
examined the alterations of form which the formulae undergo 


when we allow for changes of temperature. Weber was 
also the first to experiment on elastic after-strain. Other 
important experiments were made by different scientists, 
which disclosed a wider range of phenomena, and demanded a 
more comprehensive theory. Set was investigated by Gerstner 
(1756-1832) and Eaton Hodgkinson, while the latter physicist 
in England and Vicat (1786-1861) in France experimented 
extensively on absolute strength. Vicat boldly attacked 
the mathematical theories of flexure because they failed to 
consider shear and the time-element. As a result, a truer 
theory of flexure was soon propounded by Saint- Venant. 
Poncelet advanced the theories of resilience and cohesion. 

Gabriel Lame [94] (1795-1870) was born at Tours, and 
graduated at the Polytechnic School. He was called to Russia 
with Clapeyron and others to superintend the construction 
of bridges and roads. On his return, in 1832, he was elected 
professor of physics at the Polytechnic School. Subsequently 
he held various engineering posts and professorships in Paris. 
As engineer he took an active part in the construction of the 
first railroads in France. Lame devoted his fine mathematical 
talents mainly to mathematical physics. In four works: Legons 
sur les fonctions inverses des trans cendantes et les surfaces 
isothermes; Sur les coordonnees curvilignes et leurs diver ses 
applications; Sur la theorie analytique de la chaleur; Sur la 
theorie mathematique de I 'elasticity des corps solides (1852), 
and in various memoirs he displays fine analytical powers; but 
a certain want of physical touch sometimes reduces the value 
of his contributions to elasticity and other physical subjects. 


In considering the temperature in the interior of an ellipsoid 
under certain conditions, he employed functions analogous 
to Laplace's functions, and known by the name of "Lame's 
functions." A problem in elasticity called by Lame's name, 
viz. to investigate the conditions for equilibrium of a spherical 
elastic envelope subject to a given distribution of load on 
the bounding spherical surfaces, and the determination of 
the resulting shifts is the only completely general problem 
on elasticity which can be said to be completely solved. He 
deserves much credit for his derivation and transformation of 
the general elastic equations, and for his application of them 
to double refraction. Rectangular and triangular membranes 
were shown by him to be connected with questions in the 
theory of numbers. The field of photo-elasticity was entered 
upon by Lame, F. E. Neumann, Clerk Maxwell. Stokes, 
Wertheim, R. Clausius, Jellett, threw new light upon the 
subject of "rari-constancy" and "multi-constancy," which has 
long divided elasticians into two opposing factions. The uni- 
constant isotropy of Navier and Poisson had been questioned 
by Cauchy, and was now severely criticised by Green and 

Barre de Saint-Venant (1797-1886), ingenieur des ponts 
et chaussees, made it his life-work to render the theory of 
elasticity of practical value. The charge brought by practical 
engineers, like Vicat, against the theorists led Saint-Venant to 
place the theory in its true place as a guide to the practical man. 
Numerous errors committed by his predecessors were removed. 
He corrected the theory of flexure by the consideration of 


slide, the theory of elastic rods of double curvature by the 
introduction of the third moment, and the theory of torsion 
by the discovery of the distortion of the primitively plane 
section. His results on torsion abound in beautiful graphic 
illustrations. In case of a rod, upon the side surfaces of which 
no forces act, he showed that the problems of flexure and 
torsion can be solved, if the end- forces are distributed over the 
end-surfaces by a definite law. Clebsch, in his Lehrbuch der 
Elasticitdt, 1862, showed that this problem is reversible to the 
case of side-forces without end-forces. Clebsch [68] extended 
the research to very thin rods and to very thin plates. Saint- 
Venant considered problems arising in the scientific design of 
built-up artillery, and his solution of them differs considerably 
from Lame's solution, which was popularised by Rankine, and 
much used by gun- designers. In Saint- Venant's translation 
into French of Clebsch's Elasticitdt, he develops extensively a 
double-suffix notation for strain and stresses. Though often 
advantageous, this notation is cumbrous, and has not been 
generally adopted. Karl Pearson, professor in University 
College, London, has recently examined mathematically the 
permissible limits of the application of the ordinary theory of 
flexure of a beam. 

The mathematical theory of elasticity is still in an unsettled 
condition. Not only are scientists still divided into two schools 
of "rari-constancy" and "multi-constancy," but difference of 
opinion exists on other vital questions. Among the numerous 
modern writers on elasticity may be mentioned Emile Mathieu 
(1835-1891), professor at Besangon, Maurice Levy of Paris, 


Charles Chree, superintendent of the Kew Observatory, A. B. 
Basset, Sir William Thomson (Lord Kelvin) of Glasgow, 
J. Boussinesq of Paris, and others. Sir William Thomson 
applied the laws of elasticity of solids to the investigation 
of the earth's elasticity, which is an important element in 
the theory of ocean-tides. If the earth is a solid, then its 
elasticity co-operates with gravity in opposing deformation 
due to the attraction of the sun and moon. Laplace had 
shown how the earth would behave if it resisted deformation 
only by gravity. Lame had investigated how a solid sphere 
would change if its elasticity only came into play. Sir William 
Thomson combined the two results, and compared them with 
the actual deformation. Thomson, and afterwards G. H. 
Darwin, computed that the resistance of the earth to tidal 
deformation is nearly as great as though it were of steel. This 
conclusion has been confirmed recently by Simon Newcomb, 
from the study of the observed periodic changes in latitude. 
For an ideally rigid earth the period would be 360 days, but 
if as rigid as steel, it would be 441, the observed period being 
430 days. 

Among text-books on elasticity may be mentioned the 
works of Lame, Clebsch, Winkler, Beer, Mathieu, W. J. 
Ibbetson, and F. Neumann, edited by O. E. Meyer. 

Riemann's opinion that a science of physics only exists since 
the invention of differential equations finds corroboration 
even in this brief and fragmentary outline of the progress of 
mathematical physics. The undulatory theory of light, first 
advanced by Huygens, owes much to the power of mathematics: 


by mathematical analysis its assumptions were worked out to 
their last consequences. Thomas Young [95] (1773-1829) 
was the first to explain the principle of interference, both of 
light and sound, and the first to bring forward the idea of 
transverse vibrations in light waves. Young's explanations, 
not being verified by him by extensive numerical calculations, 
attracted little notice, and it was not until Augustin Fresnel 
(1788-1827) applied mathematical analysis to a much greater 
extent than Young had done, that the undulatory theory 
began to carry conviction. Some of Fresnel's mathematical 
assumptions were not satisfactory; hence Laplace, Poisson, 
and others belonging to the strictly mathematical school, at 
first disdained to consider the theory. By their opposition 
Fresnel was spurred to greater exertion. Arago was the 
first great convert made by Fresnel. When polarisation and 
double refraction were explained by Young and Fresnel, then 
Laplace was at last won over. Poisson drew from Fresnel's 
formulae the seemingly paradoxical deduction that a small 
circular disc, illuminated by a luminous point, must cast a 
shadow with a bright spot in the centre. But this was found 
to be in accordance with fact. The theory was taken up 
by another great mathematician, Hamilton, who from his 
formulae predicted conical refraction, verified experimentally 
by Lloyd. These predictions do not prove, however, that 
Fresnel's formulae are correct, for these prophecies might 
have been made by other forms of the wave-theory The 
theory was placed on a sounder dynamical basis by the 
writings of Cauchy, Biot, Green, C. Neumann, Kirchhoff, 


McCullagh, Stokes, Saint- Venant, Sarrau, Lorenz, and Sir 
William Thomson. In the wave-theory, as taught by Green 
and others, the luminiferous ether was an incompressible 
elastic solid, for the reason that fluids could not propagate 
transverse vibrations. But, according to Green, such an 
elastic solid would transmit a longitudinal disturbance with 
infinite velocity. Stokes remarked, however, that the ether 
might act like a fluid in case of finite disturbances, and like an 
elastic solid in case of the infinitesimal disturbances in light 

Fresnel postulated the density of ether to be different in 
different media, but the elasticity the same, while C. Neumann 
and McCullagh assume the density uniform and the elasticity 
different in all substances. On the latter assumption the 
direction of vibration lies in the plane of polarisation, and not 
perpendicular to it, as in the theory of Fresnel. 

While the above writers endeavoured to explain all optical 
properties of a medium on the supposition that they arise 
entirely from difference in rigidity or density of the ether 
in the medium, there is another school advancing theories 
in which the mutual action between the molecules of the 
body and the ether is considered the main cause of refrac- 
tion and dispersion. [100] The chief workers in this field are 
J. Boussinesq, W. Sellmeyer, Helmholtz, E. Lommel, E. Ket- 
teler, W. Voigt, and Sir William Thomson in his lectures 
delivered at the Johns Hopkins University in 1884. Neither 
this nor the first-named school succeeded in explaining all 
the phenomena. A third school was founded by Maxwell. 


He proposed the electro-magnetic theory, which has received 
extensive development recently. It will be mentioned again 
later. According to Maxwell's theory, the direction of vi- 
bration does not lie exclusively in the plane of polarisation, 
nor in a plane perpendicular to it, but something occurs in 
both planes — a magnetic vibration in one, and an electric in 
the other. Fitzgerald and Trouton in Dublin verified this 
conclusion of Maxwell by experiments on electro-magnetic 

Of recent mathematical and experimental contributions to 
optics, mention must be made of H. A. Rowland's theory 
of concave gratings, and of A. A. Michelson's work on 
interference, and his application of interference methods to 
astronomical measurements. 

In electricity the mathematical theory and the measure- 
ments of Henry Cavendish (1731-1810), and in magnetism 
the measurements of Charles Augustin Coulomb (1736- 
1806), became the foundations for a system of measurement. 
For electro-magnetism the same thing was done by Andre 
Marie Ampere (1775-1836). The first complete method 
of measurement was the system of absolute measurements 
of terrestrial magnetism introduced by Gauss and Wilhelm 
Weber (1804-1891) and afterwards extended by Wilhelm 
Weber and F. Kohlrausch to electro-magnetism and electro- 
statics. In 1861 the British Association and the Royal Society 
appointed a special commission with Sir William Thomson 
at the head, to consider the unit of electrical resistance. The 
commission recommended a unit in principle like W. Weber's, 


but greater than Weber's by a factor of 10 7 . [101] The discus- 
sions and labours on this subject continued for twenty years, 
until in 1881 a general agreement was reached at an electrical 
congress in Paris. 

A function of fundamental importance in the mathematical 
theories of electricity and magnetism is the "potential." It was 
first used by Lagrange in the determination of gravitational 
attractions in 1773. Soon after, Laplace gave the celebrated 
differential equation, 

d 2 V d 2 V d 2 V _ 
dx 2 dy 2 dz 2 

which was extended by Poisson by writing — ink in place 
of zero in the right-hand member of the equation, so that 
it applies not only to a point external to the attracting 
mass, but to any point whatever. The first to apply the 
potential function to other than gravitation problems was 
George Green (1793-1841). He introduced it into the 
mathematical theory of electricity and magnetism. Green 
was a self-educated man who started out as a baker, and 
at his death was fellow of Caius College, Cambridge. In 
1828 he published by subscription at Nottingham a paper 
entitled Essay on the application of mathematical analysis to 
the theory of electricity and magnetism. It escaped the notice 
even of English mathematicians until 1846, when Sir William 
Thomson had it reprinted in Crelle 's Journal, vols. xliv. and 
xlv. It contained what is now known as "Green's theorem" for 
the treatment of potential. Meanwhile all of Green's general 
theorems had been re-discovered by Sir William Thomson, 


Chasles, Sturm, and Gauss. The term potential function is 
due to Green. Hamilton used the word force-function, while 
Gauss, who about 1840 secured the general adoption of the 
function, called it simply potential. 

Large contributions to electricity and magnetism have been 
made by William Thomson. He was born in 1824 at Belfast, 
Ireland, but is of Scotch descent. He and his brother James 
studied in Glasgow. From there he entered Cambridge, and 
was graduated as Second Wrangler in 1845. William Thomson, 
Sylvester, Maxwell, Clifford, and J. J. Thomson are a group 
of great men who were Second Wranglers at Cambridge. At 
the age of twenty-two W. Thomson was elected professor of 
natural philosophy in the University of Glasgow, a position 
which he has held ever since. For his brilliant mathematical 
and physical achievements he was knighted, and in 1892 was 
made Lord Kelvin. His researches on the theory of potential 
are epoch-making. What is called "Dirichlet's principle" 
was discovered by him in 1848, somewhat earlier than by 
Dirichlet. We owe to Sir William Thomson new synthetical 
methods of great elegance, viz. the theory of electric images 
and the method of electric inversion founded thereon. By 
them he determined the distribution of electricity on a bowl, 
a problem previously considered insolvable. The distribution 
of static electricity on conductors had been studied before 
this mainly by Poisson and Plana. In 1845 F. E. Neumann of 
Konigsberg developed from the experimental laws of Lenz the 
mathematical theory of magneto-electric induction. In 1855 
W. Thomson predicted by mathematical analysis that the 


discharge of a Leyden jar through a linear conductor would 
in certain cases consist of a series of decaying oscillations. 
This was first established experimentally by Joseph Henry of 
Washington. William Thomson worked out the electro-static 
induction in submarine cables. The subject of the screening 
effect against induction, due to sheets of different metals, 
was worked out mathematically by Horace Lamb and also 
by Charles Niven. W. Weber's chief researches were on 
electro-dynamics. Helmholtz in 1851 gave the mathematical 
theory of the course of induced currents in various cases. 
Gustav Robert Kirchhoff [97] (1824-1887) investigated 
the distribution of a current over a flat conductor, and also 
the strength of current in each branch of a network of linear 

The entire subject of electro-magnetism was revolutionised 
by James Clerk Maxwell (1831-1879). He was born 
near Edinburgh, entered the University of Edinburgh, and 
became a pupil of Kelland and Forbes. In 1850 he went to 
Trinity College, Cambridge, and came out Second Wrangler, 
E. Routh being Senior Wrangler. Maxwell then became 
lecturer at Cambridge, in 1856 professor at Aberdeen, and 
in 1860 professor at King's College, London. In 1865 he 
retired to private life until 1871, when he became professor 
of physics at Cambridge. Maxwell not only translated into 
mathematical language the experimental results of Faraday, 
but established the electro-magnetic theory of light, since 
verified experimentally by Hertz. His first researches thereon 
were published in 1864. In 1871 appeared his great Treatise 


on Electricity and Magnetism. He constructed the electro- 
magnetic theory from general equations, which are established 
upon purely dynamical principles, and which determine the 
state of the electric field. It is a mathematical discussion of the 
stresses and strains in a dielectric medium subjected to electro- 
magnetic forces. The electro-magnetic theory has received 
developments from Lord Rayleigh, J. J. Thomson, H. A. 
Rowland, R. T. Glazebrook, H. Helmholtz, L. Boltzmann, 
O. Heaviside, J. H. Poynting, and others. Hermann 
von Helmholtz turned his attention to this part of the 
subject in 1871. He was born in 1821 at Potsdam, studied at 
the University of Berlin, and published in 1847 his pamphlet 
Ueber die Erhaltung der Kraft. He became teacher of anatomy 
in the Academy of Art in Berlin. He was elected professor 
of physiology at Konigsberg in 1849, at Bonn in 1855, at 
Heidelberg in 1858. It was at Heidelberg that he produced 
his work on Tonempfindung. In 1871 he accepted the chair 
of physics at the University of Berlin. From this time on 
he has been engaged chiefly on inquiries in electricity and 
hydrodynamics. Helmholtz aimed to determine in what 
direction experiments should be made to decide between 
the theories of W. Weber, F. E. Neumann, Riemann, and 
Clausius, who had attempted to explain electro-dynamic 
phenomena by the assumption of forces acting at a distance 
between two portions of the hypothetical electrical fluid,— 
the intensity being dependent not only on the distance, but 
also on the velocity and acceleration, — and the theory of 
Faraday and Maxwell, which discarded action at a distance 


and assumed stresses and strains in the dielectric. His 
experiments favoured the British theory. He wrote on 
abnormal dispersion, and created analogies between electro- 
dynamics and hydrodynamics. Lord Rayleigh compared 
electro-magnetic problems with their mechanical analogues, 
gave a dynamical theory of diffraction, and applied Laplace's 
coefficients to the theory of radiation. Rowland made some 
emendations on Stokes' paper on diffraction and considered 
the propagation of an arbitrary electro-magnetic disturbance 
and spherical waves of light. Electro-magnetic induction has 
been investigated mathematically by Oliver Heaviside, and 
he showed that in a cable it is an actual benefit. Heaviside 
and Poynting have reached remarkable mathematical results 
in their interpretation and development of Maxwell's theory. 
Most of Heaviside's papers have been published since 1882; 
they cover a wide field. 

One part of the theory of capillary attraction, left defective 
by Laplace, namely, the action of a solid upon a liquid, and 
the mutual action between two liquids, was made dynamically 
perfect by Gauss. He stated the rule for angles of contact 
between liquids and solids. A similar rule for liquids was 
established by Ernst Franz Neumann. Chief among recent 
workers on the mathematical theory of capillarity are Lord 
Rayleigh and E. Mathieu. 

The great principle of the conservation of energy was estab- 
lished by Robert Mayer (1814-1878), a physician in Heil- 
bronn, and again independently by Colding of Copenhagen, 
Joule, and Helmholtz. James Prescott Joule (1818-1889) 


determined experimentally the mechanical equivalent of heat. 
Helmholtz in 1847 applied the conceptions of the transfor- 
mation and conservation of energy to the various branches 
of physics, and thereby linked together many well-known 
phenomena. These labours led to the abandonment of the 
corpuscular theory of heat. The mathematical treatment of 
thermic problems was demanded by practical considerations. 
Thermodynamics grew out of the attempt to determine math- 
ematically how much work can be gotten out of a steam engine. 
Sadi-Carnot, an adherent of the corpuscular theory, gave 
the first impulse to this. The principle known by his name 
was published in 1824. Though the importance of his work 
was emphasised by B. P. E. Clapeyron, it did not meet with 
general recognition until it was brought forward by William 
Thomson. The latter pointed out the necessity of modifying 
Carnot's reasoning so as to bring it into accord with the 
new theory of heat. William Thomson showed in 1848 that 
Carnot's principle led to the conception of an absolute scale 
of temperature. In 1849 he published "an account of Carnot's 
theory of the motive power of heat, with numerical results 
deduced from Regnault's experiments." In February, 1850, 
Rudolph Clausius (1822-1888), then in Zurich (afterwards 
professor in Bonn), communicated to the Berlin Academy 
a paper on the same subject which contains the Protean 
second law of thermodynamics. In the same month William 
John M. Rankine (1820-1872), professor of engineering 
and mechanics at Glasgow, read before the Royal Society of 
Edinburgh a paper in which he declares the nature of heat 


to consist in the rotational motion of molecules, and arrives 
at some of the results reached previously by Clausius. He 
does not mention the second law of thermodynamics, but in 
a subsequent paper he declares that it could be derived from 
equations contained in his first paper. His proof of the second 
law is not free from objections. In March, 1851, appeared 
a paper of William Thomson which contained a perfectly 
rigorous proof of the second law. He obtained it before he had 
seen the researches of Clausius. The statement of this law, 
as given by Clausius, has been much criticised, particularly 
by Rankine, Theodor Wand, P. G. Tait, and Tolver Preston. 
Repeated efforts to deduce it from general mechanical princi- 
ples have remained fruitless. The science of thermodynamics 
was developed with great success by Thomson, Clausius, and 
Rankine. As early as 1852 Thomson discovered the law of 
the dissipation of energy, deduced at a later period also by 
Clausius. The latter designated the non-transformable energy 
by the name entropy, and then stated that the entropy of 
the universe tends toward a maximum. For entropy Rankine 
used the term thermodynamic function. Thermodynamic 
investigations have been carried on also by G. Ad. Hirn of 
Colmar, and Helmholtz (monocyclic and polycyclic systems). 
Valuable graphic methods for the study of thermodynamic 
relations were devised in 1873-1878 by J. Willard Gibbs of 
Yale College. Gibbs first gives an account of the advantages 
of using various pairs of the five fundamental thermodynamic 
quantities for graphical representation, then discusses the 
entropy-temperature and entropy-volume diagrams, and the 


volume-energy-entropy surface (described in Maxwell's The- 
ory of Heat). Gibbs formulated the energy-entropy criterion of 
equilibrium and stability, and expressed it in a form applicable 
to complicated problems of dissociation. Important works on 
thermodynamics have been prepared by Clausius in 1875, by 
R. Ruhlmann in 1875, and by Poincare in 1892. 

In the study of the law of dissipation of energy and the 
principle of least action, mathematics and metaphysics met 
on common ground. The doctrine of least action was first 
propounded by Maupertius in 1744. Two years later he 
proclaimed it to be a universal law of nature, and the first 
scientific proof of the existence of God. It was weakly 
supported by him, violently attacked by Konig of Leipzig, 
and keenly defended by Euler. Lagrange's conception of 
the principle of least action became the mother of analytic 
mechanics, but his statement of it was inaccurate, as has 
been remarked by Josef Bertrand in the third edition of the 
Mecanique Analytique. The form of the principle of least 
action, as it now exists, was given by Hamilton, and was 
extended to electro-dynamics by F. E. Neumann, Clausius, 
Maxwell, and Helmholtz. To subordinate the principle to 
all reversible processes, Helmholtz introduced into it the 
conception of the "kinetic potential." In this form the 
principle has universal validity. 

An offshoot of the mechanical theory of heat is the modern 
kinetic theory of gases, developed mathematically by Clausius, 
Maxwell, Ludwig Boltzmann of Munich, and others. The first 
suggestions of a kinetic theory of matter go back as far as the 


time of the Greeks. The earliest work to be mentioned here is 
that of Daniel Bernoulli, 1738. He attributed to gas- molecules 
great velocity, explained the pressure of a gas by molecular 
bombardment, and deduced Boyle's law as a consequence of 
his assumptions. Over a century later his ideas were taken 
up by Joule (in 1846), A. K. Kronig (in 1856), and Clausius 
(in 1857). Joule dropped his speculations on this subject when 
he began his experimental work on heat. Kronig explained by 
the kinetic theory the fact determined experimentally by Joule 
that the internal energy of a gas is not altered by expansion 
when no external work is done. Clausius took an important 
step in supposing that molecules may have rotary motion, and 
that atoms in a molecule may move relatively to each other. He 
assumed that the force acting between molecules is a function 
of their distances, that temperature depends solely upon the 
kinetic energy of molecular motions, and that the number of 
molecules which at any moment are so near to each other 
that they perceptibly influence each other is comparatively 
so small that it may be neglected. He calculated the average 
velocities of molecules, and explained evaporation. Objections 
to his theory, raised by Buy's-Ballot and by Jochmann, were 
satisfactorily answered by Clausius and Maxwell, except in one 
case where an additional hypothesis had to be made. Maxwell 
proposed to himself the problem to determine the average 
number of molecules, the velocities of which lie between given 
limits. His expression therefor constitutes the important 
law of distribution of velocities named after him. By this 
law the distribution of molecules according to their velocities 


is determined by the same formula (given in the theory 
of probability) as the distribution of empirical observations 
according to the magnitude of their errors. The average 
molecular velocity as deduced by Maxwell differs from that of 
Clausius by a constant factor. Maxwell's first deduction of 
this average from his law of distribution was not rigorous. A 
sound derivation was given by O. E. Meyer in 1866. Maxwell 
predicted that so long as Boyle's law is true, the coefficient of 
viscosity and the coefficient of thermal conductivity remain 
independent of the pressure. His deduction that the coefficient 
of viscosity should be proportional to the square root of the 
absolute temperature appeared to be at variance with results 
obtained from pendulum experiments. This induced him 
to alter the very foundation of his kinetic theory of gases 
by assuming between the molecules a repelling force varying 
inversely as the fifth power of their distances. The founders 
of the kinetic theory had assumed the molecules of a gas to be 
hard elastic spheres; but Maxwell, in his second presentation 
of the theory in 1866, went on the assumption that the 
molecules behave like centres of forces. He demonstrated 
anew the law of distribution of velocities; but the proof had a 
flaw in argument, pointed out by Boltzmann, and recognised 
by Maxwell, who adopted a somewhat different form of the 
distributive function in a paper of 1879, intended to explain 
mathematically the effects observed in Crookes' radiometer. 
Boltzmann gave a rigorous general proof of Maxwell's law of 
the distribution of velocities. 

None of the fundamental assumptions in the kinetic theory 


of gases leads by the laws of probability to results in very 
close agreement with observation. Boltzmann tried to estab- 
lish kinetic theories of gases by assuming the forces between 
molecules to act according to different laws from those pre- 
viously assumed. Clausius, Maxwell, and their predecessors 
took the mutual action of molecules in collision as repulsive, 
but Boltzmann assumed that they may be attractive. Exper- 
iments of Joule and Lord Kelvin seem to support the latter 

Among the latest researches on the kinetic theory is 
Lord Kelvin's disproof of a general theorem of Maxwell 
and Boltzmann, asserting that the average kinetic energy of 
two given portions of a system must be in the ratio of the 
number of degrees of freedom of those portions. 


PAGE 15. The new Akhmim papyrus, written in Greek, is probably 
the copy of an older papyrus, antedating Heron's works, and is the oldest 
extant text-book on practical Greek arithmetic. It contains, besides 
arithmetical examples, a table for finding "unit-fractions," identical in 
scope with that of Ahmes, and, like Ahmes's, without a clue as to its 
mode of construction. See Biblioth. Math., 1893, p. 79-89. The 
papyrus is edited by J. Baillet [Memoires publies par les membres de 
la mission archeologique frangaise au Caire, T. IX., I 1 ' fascicule, Paris, 
1892, p. 1-88). 

PAGE 45. Chasles's or Simson's definition of a Porism is preferable 
to Proclus's, given in the text. See Gow, p. 217-221. 

PAGE 132. Nasir Eddin for the first time elaborated trigonometry 
independently of astronomy and to such great perfection that, had his 
work been known, Europeans of the 15th century might have spared 
their labours. See Biblioth. Math., 1893, p. 6. 

Page 134. This law of sines was probably known before Gabir ben 
Aflah to Tabit ben Korra and others. See Biblioth. Math., 1893, p. 7. 

Page 145. Athelard was probably not the first to translate Euclid's 
Elements from the Arabic. See M. Cantor's Vorlesungen, Vol. II., 
p. 91, 92. 

Page 279. G. Enestrom argues that Taylor and not Nicole is the 
real inventor of finite differences. See Biblioth. Math., 1893, p. 91. 

Page 290. An earlier publication in which 3.14159 ... is designated 
by it, is W. Jones's Synopsis palmariorum matheseos, London, 1706, 
p. 243, 263 et seq. See Biblioth. Math., 1894, p. 106. 

PAGE 391. Before Gauss a theorem on convergence, usually 
attributed to Cauchy, was given by Maclaurin (Fluxions, § 350). A rule 
of convergence was deduced also by Stirling. See Bull. N. Y. Math. 
Soc, Vol. Ill, p. 186. 

Page 418. The surface of a solid with p holes was considered before 
Clifford by Tonelli, and was probably used by Riemann himself. See 
Math. Annalen, Vol. 45, p. 142. 

Page 421. As early as 1835, Lobachevsky showed in a memoir the 
necessity of distinguishing between continuity and differentiability. See 
G. B. Halsted's transl. of A. Vasiliev's Address on Lobachevsky, p. 23. 



Recent deaths. Johann Rudolf Wolf, Dec. 6, 1893; Heinrich Hertz, 
Jan. 1, 1894; Eugene Catalan, Feb. 14, 1894; Hermann von Helmholtz, 
Sept. 8, 1894; Arthur Cayley, Jan. 26, 1895. 


Abacists, 146 

Abacus, 8, 13, 73, 91, 94, 138, 

141, 146, 150 
Abbatt, 389 
Abel, 405 

ref. to, 170, 325, 339, 364, 382, 

391, 393, 408, 412, 433 
Abel's theorem, 410 

Abelian functions, 340, 364, 382, 
403, 405, 407, 411, 414-417, 

Abelian integrals, 408, 442 

Absolute geometry, 351 

Absolutely convergent series, 390, 

392, 394 
Abul Gud, 128 

ref. to, 131 
Abul Hasan, 133 
Abul Wefa, 127 

ref. to, 129, 130 
Achilles and tortoise, paradox of, 

Acoustics, 304, 314, 323, 450 

varying, 340, 371, 442 
Action, least, 294, 426, 468 
Adams, 438 

ref. to, 249 
Addition theorem of elliptic 
integrals, 293, 408, 462 
Adrain, 322 
iEquipollences, 375 
Agnesi, 303 
Agrimensores, 92 
Ahmes, 11-16 

ref. to, 19, 20, 61, 86, 151 

Airy, 438 

ref. to, 447 
Al Battani, 126 

ref. to, 127, 145 
Albertus Magnus, 155 
Albiruni, 128 

ref. to, 118, 120 
Alcuin, 137 

Alembert, D', See D'Alembert 
Alexandrian School 

(first), 39-62 

(second), 62-71 
Alfonso's tables, 147 
Algebra, See Notation 

Arabic, 124, 128, 133 

Beginnings in Egypt, 16 

Diophantus, 86-89 

early Greek, 84 

Hindoo, 107-110 

Lagrange, 310 

Middle Ages, 155, 157 

origin of terms, 124, 133 

Peacock, 331 

recent, 367-385 

Renaissance, 162, 165-174, 177 

seventeenth century, 192, 218, 
Algebraic functions, 403 

integrals, 439 

Middle Ages, 146, 149 

origin of term, 123 
Al Haitam, 133 

ref. to, 130 
Al Hayyami, 129 

ref. to, 131 




Al Hazin, 130 
Al Hogendi, 128 
Al Karhi, 128, 131 
Al Kaschi, 132 
Al Kuhi, 128 

ref. to, 130 
Allegret, 440 
Allman, ix, 42 
Al Madshriti, 133 
Almagest, 65-67 

ref. to, 121, 126, 147, 156, 158, 
Al Mahani, 130 
Alphonso's tables, 147 
Al Sagani, 128 
Alternate numbers, 375 
Ampere, 460 

ref. to, 421 
Amyclas, 38 

(in synthetic geometry), 35, 45 

Descartes', 216 

modern, 386-390 
Analysis situs, 262, 367 
Analytic geometry, 215-219, 222, 

224, 279, 334, 358-366 
Analytical Society (in 
Cambridge), 330 
Anaxagoras, 21 

ref. to, 32 
Anaximander, 20 
Anaximenes, 21 
Angeli, 216 
Anger, 437 
Anharmonic ratio, 207, 342, 346, 

Anthology, Palatine, 84, 138 
Antiphon, 30 

ref. to, 30 
Apices of Boethius, 94 

ref. to, 73, 119, 138, 146, 150 
Apollonian Problem, 57, 179, 218 
Apollonius, 51-58 

ref. to, 41, 42, 45, 62, 70, 76, 
90, 121, 125, 133, 163, 178 
Appel, 403 
Applied mathematics, See 

Astronomy, Mechanics, 
Arabic manuscripts, 144-148 
Arabic numerals and notation, 3, 
84, 100, 118, 129, 147-149, 
Arabs, 116-135 
Arago, xii, 387, 458 
Arbogaste, 302 
Archimedes, 46-52 

ref. to, 2, 40, 42, 44, 51, 53, 56, 
57, 62, 71, 75, 85, 90, 104, 
121, 125, 163, 167, 196, 201, 
Archytas, 25 

ref. to, 33, 35, 37, 49 
Areas, conservation of, 294 
Arenarius, 76 
Argand, 370 

ref. to, 307 
Aristasus, 38 
ref. to, 53 
Aristotle, 39 

ref. to, 9, 18, 31, 49, 71, 78, 144 
Arithmetic, See Numbers, 
Arabic, 122 
Euclid, 44, 81 
Greek, 72-88 
Hindoo, 103-106 
Middle Ages, 137, 142, 146, 

150, 154, 156 
Platonists, 33 



Pythagoreans, 23, 77-81 

Renaissance, 174, 175, 184-186 
Arithmetical machine, 255, 330 
Arithmetical triangle, 228 
Armemante, 364 
Arneth,, x 
Aronhold, 381 
Aryabhatta, 100 

ref. to, 102, 106, 114 
Aschieri, 356 

Assumption, tentative, See 
Regula falsa, 87, 107 
Astrology, 180 
Astronomy, See Mechanics 

Arabic, 115, 117, 121, 134 

Babylonian, 9 

Egyptian, 10 

Greek, 20, 27, 36, 45, 58, 64 

Hindoo, 99 

Middle Ages, 147 

more recent researches, 294, 
298, 304, 315-319, 426, 

Newton, 246-251 
Athelard of Bath, 145 

ref. to, 156 
Athenaeus, 37 
Atomic theory, 446 
Attalus, 53 
Attraction, See Gravitation, 

Ellipsoid, 322 
August, 344 

Ausdehnungslehre, 373, 374, 441 
Axioms (of geometry), 34, 42, 43, 
327, 350, 367 

Babbage, 330, 415 
Babylonians, 5-9 
ref. to, 21, 59 
Bachet de Meziriac, See Meziriac 

Bachmann, 433 

ref. to, 426 
Bacon, R., 156 
Baker, Th, 131 
Ball, Sir R. S., 441 
Ball, W. W. R., xi, 253 
Ballistic curve, 324 
Baltzer, R., 366 

ref. to, 352, 379 
Barbier, 397 
Barrow, 230 

ref. to, 201, 234, 235, 257, 264 
Basset, 444, 446 
Battaglini, 356 
Bauer, xiii 
Baumgart, xii 
Baune, De, See De Baune 
Bayes, 396 
Beaumont, xii 
Bede, the Venerable, 137 
Beer, 457 
Beha Eddin, 132 
Bellavitis, 375 

ref. to, 349, 354, 369 
Beltrami, 354, 355 

ref. to, 367 
Ben Junus, 133 
Berkeley, 274 
Bernelinus, 141 
Bernoulli's theorem, 276 
Bernoulli, Daniel, 276 

ref. to, 297, 305, 450, 469 
Bernoulli, James (born 1654), 
275, 276 

ref. to, 212, 262, 267, 291 
Bernoulli, James (born 1758), 

278, 415, 451 
Bernoulli, John (born 1667), 276 

ref. to, 262, 267, 270, 272, 275, 
282, 291, 415 



Bernoulli, John (born 1710), 278 
Bernoulli, John (born 1744), 278 
Bernoulli, Nicolaus (born 1687), 

278, 291, 314 
Bernoulli, Nicolaus (born 1695), 

Bernoullis, genealogical table of, 

Bertini, 356 
Bertrand, 393, 396, 399, 440, 443, 

444, 468 
Bessel, 435-437 

ref. to, 353, 360, 409 
Bessel's functions, 436 
Bessy, 210 
Beta function, 289 
Betti, 412 
Beyer, 186 
Bezout, 302 

ref. to, 290, 307 
Bezout's method of elimination, 

302, 385 
Bhaskara, 100 

ref. to, 106-110, 112, 177 
Bianchi, 382 
Billingsley, 160 
Binet, 378, 451 
Binomial formula, 226, 228, 234, 

292, 405 
Biot, 320, 335, 459 
Biquadratic equation, 130, 169, 

Biquadratic residues, 426 
Biquaternions, 441 
Bjerknes, C. A., xiv, 417, 447 
Bobillier, 358 
Bocher, xv 
Bode, 398 
Boethius, 94 

ref. to, 73, 83, 119, 137, 140, 

Bois-Reymond, P. du, xiv, 

393-396, 422 
Boltzmann, 464, 470 
Bolyai, Johann, 351 

ref. to, 338 
Bolyai, Wolfgang, 351 

ref. to, 338, 424 
Bolza, 408 
Bombelli, 169 

ref. to, 177 
Bonnet, O., 366 

ref. to, 393, 399 
Boole, 400 

ref. to, 339, 379, 396, 398, 404 
Booth, 362 
Borchardt, 414 
Bouniakowsky, 426 
Bouquet, 402 

ref. to, 404, 413 
Bour, 397, 440 
Boussinesq, 447, 457, 459 
Bowditch, 320, 376 
Boyle's law, 469 
Brachistochrone (line of swiftest 

descent), 272, 276 
Bradwardine, 156 

ref. to, 163 
Brahe, Tycho, 127, 161, 195 
Brahmagupta, 100 

ref. to, 106, 110, 113, 117 
Bredon, 157 

Bretschneider, ix, 113, 373 
Brianchion, 206, 335, 336 
Briggs, 190 

Brill, A., 346, 363, 415 
Brill, L., 357 
Bring, 383 
Brioschi, 382 

ref. to, 379, 385, 389, 408, 412, 



Briot, 401 

ref. to, 404, 413 
Brouncker, 229 
Bruno, Faa de, 382 
Bruns, 440 

Bryson of Heraclea, 30 
Buchheim, 441 

ref. to, 357 
Buckley, 185 
Budan, 328 
Buddha, 103 
Buffon, 397 
Bungus, 180 
Burgi, 186 

ref. to, 192 
Burkhardt, H., xiii, 382 
Burkhardt, J. K., 320 
Burmester, 350 
Busche, 426 
Buteo, 179 
Buy's-Ballot, 469 
Byrgius, See Burgi 

Cassar, Julius, 93 
Calculating machines, 255, 330 
Calculation, origin of word, 91 
Calculus, See Differential 

of operations, 340 
of variations, 287, 289, 304, 

309, 344, 382, 388-389, 415, 

Calendar, 9, 93, 164, 179, 315 
Callisthenes, 9 
Canon paschalis, 91 
Cantor, C, 396, 422, 434 
Cantor, M., ix, x, 129 
Capelli, 384 
Capillarity, 323, 426, 452, 465 

Caporali, 365 
Cardan, 167 

ref. to, 172, 176, 181, 185 
Carll, 390 
Carnot, Lazare, 335, 336 

ref. to, 64, 274, 342 
Carnot, Sadi, 466 
Casey, 365 
Cassini, D, 299 
Cassiodorius, 95, 137 
Casting out the 9's, 105, 123 
Catalan, E., 378 
Cataldi, 185 
Catenary, 222, 272, 276 
Cattle-problem, 85 
Cauchy, 386-388 

ref. to, 282, 287, 307, 375, 378, 
382, 385, 391, 394, 395, 398, 
399, 402, 406-408, 411, 415, 
422, 430, 446, 452, 453, 455, 
Caustics, 276, 280 
Cavalieri, 197 

ref. to, 194, 225, 257 
Cavendish, 460 
Cayley, xii, xv, 379 

ref. to, 339, 344, 346, 356, 359, 
362, 365, 372, 377, 385, 404, 
413, 414 

of gravity, 205, 222 

of oscillation, 222, 282 
Centres of osculation, 56 
Centrifugal force, 212, 223, 248 
Ceulen, van, See Ludolph 
Ceva, 338 
Chapman, 378 

Characteristics, method of, 346 
Chasles, x, 345-347 

ref. to, 45, 54, 56, 60, 200, 337, 



342, 356, 362, 365, 440 

Chauvenet, 440 
Chess, 107 
Cheyne, 239 
Chinese, 21 
Chladni's figures, 450 
Chree, 446, 457 
Christoffel, 378, 382 
Circle, 21, 27-32, 35, 47, 59, 178, 

degrees of, 8, 315 

division of, 384, 426 
Circle-squarers, 2, 21, 220, 369 
Cissoid, 58, 222 
Clairaut, 298-300 

ref. to, 284, 293, 297, 305 
Clapeyron, 466 
Clarke, 397 
Clausius, 466 

ref. to, 455, 464, 467-470 
Clavius, 180 

ref. to, 179 
Clebsch, 363, 364 

ref. to, xiii, 345, 360, 367, 376, 
381, 382, 389, 397, 398, 418, 
445, 447, 456-457 
Clifford, 355, 356 

ref. to, 346, 372, 377, 418, 441, 
Co-ordinates, 215, 343, 359, 366, 

first use of term, 263 
Cockle, 367 
Colburn, Z, 210 
Colding, 465 
Cole, 384 
Colebrooke, 101 
Colla, 166, 168 

Collins, 235, 260, 264, 265, 268, 

Colson, 237 

Combinatorial School, 288, 390 
Commandinus, 177 
Commercium epistolicum, 239, 

Complex of lines, 360 
Complex quantities, See 

Imaginaries, 340, 370 
Computus, 137, 138 
Comte, x 

Concentric spheres of Eudoxus, 37 
Conchoid, 58 

Condensation of singularities, 422 
Conform representation of 

surfaces, 420 
Congruencies, theory of, 425 
Congruency of lines, 359 
Conic sections, See Geometry 

Arabs, 117, 130 

Greek, 36, 38, 45, 46, 52-57, 63 

Kepler, 195 

more recent researches, 
204-206, 224 

Renaissance, 178 
Conon, 46 

ref. to, 48 

of vis viva, 223 

of areas, 294 

of energy, 463, 465 
Continued fractions, 185, 229, 

293, 314 
Continuity, 197, 224, 263, 341, 

388, 419, 434 
Contracted vein, 448 
Contravariants, 381 
Convergence of series, 390-396 
Copernican System, 161 
Copernicus, 65, 161 
Correspondence, principle of, 341, 




Cosine, 191 

Coss, term for algebra, 176 
Cotangent, 163, 191 
Cotes, 281 

ref. to, 283 
Coulomb, 460 
Cournot, 396 
Cousinery, 348 
Covariants, 381, 412, 430 
Cox, 357 
Craig, J., 262 

Craig, T., 357, 404, 414, 446 
Cramer, 252 
Crelle, 405 

ref. to, 406 
Crelle's Journal, 344 
Cremona, 348 

ref. to, 339, 342-344, 349, 365 
Cridhara, 100 

Criteria of convergence, 389-395 
Crofton, 397 
Crozet, 336 
Ctesibius, 59 

Cube numbers, 83, 128, 209 
Cube, duplication of, See 

Duplication of the cube 
Cubic curves, 252, 298, 345 
Cubic equations, See Algebra, 

129, 130, 164-168, 172, 176, 
Cubic residues, 427 
Culmann, 348, 349 
Curtze, M, 348 
Curvature, measure of, 365 
Curve of swiftest descent, 272, 277 
Curves, See Cubic curves, 

Rectification, Geometry, 
Conic sections 

osculating, 263 

quadrature of, 48, 57, 205, 221, 

224, 235, 256 
theory of, 263, 279, 281, 283, 

340, 373 
Cusanus, 178 
Cyclic method, 111, 112 
Cycloid, 199, 201, 205, 217, 221, 

222, 262, 272, 279 
Cyzicenus, 38 
Czuber, 396 

D'Alembert, 295-298 

ref. to, 295, 300, 304, 308, 
312-315, 450 
D'Alembert's principle, 295 
Damascius, 71 

ref. to, 44, 121 
Darboux, xiv, 365, 400, 404, 421, 

Darwin, 438 

ref. to, 448, 457 
Data (Euclid's), 44 
Davis, E. W., 357 
Davis, W. M., 449 
De Baune, 219 

ref. to, 215, 259, 261 
Decimal fractions, 184-187 
Decimal point, 187 
Dedekind, 433 

ref. to, 417, 422, 434 
Dee, 160 

Deficiency of curves, 363 
Definite integrals, 196, 390, 395, 

397, 409, 422 

seeDinostratus, 36 
De Lahire, 332, 337 
Delambre, 427 
Delaunay, 438 

ref. to, 389 



Delian problem, See Duplication 

of the cube 
Del Pezzo, 356 
Democritus, 32 

ref. to, 17 
De Moivre, 279, 281, 284 
De Morgan, 368 

ref. to, xi, 1, 2, 81, 111, 187, 

239, 267, 271, 303, 322, 331, 

339, 388, 392, 396, 415 
De Paolis, 356 
Derivatives, method of, 313 
Desargues, 206 

ref. to, 203, 214, 280, 332, 337 
Desboves, 443 
Descartes, 213-220 

ref. to, 4, 55, 69, 131, 194, 201, 

202, 220, 222, 224, 252, 256, 

259, 279, 369 
rule of signs, 217, 224 
Descriptive geometry, 333-336, 

Determinants, 263, 308, 323, 364, 

378, 389, 423 
Devanagari-numerals, 119 
Dialytic method of elimination, 

Differences, finite, See Finite 

Differential calculus, See 

Bernoullis, Euler, Lagrange, 

Laplace, etc, 233, 257-264, 

alleged invention by Pascal, 

controversy between Newton 

and Leibniz, 264-270 
philosophy of, 274, 298, 301, 

312, 336, 388 
Differential equations, 277, 293, 

308, 323, 365, 371, 373, 388, 

Differential invariants, 381 
Dingeldey, 367 
Dini, 393 

ref. to, 422 
Dinostratus, 36 

ref. to, 28 
Diocles, 58 
Diodorus, 11, 46, 67 
Diogenes Laertius, 19, 37 
Dionysodorus, 62 
Diophantus, 85-89 

ref. to, 63, 70, 99, 107, 110, 

111, 121, 123, 124, 127, 128, 

208, 434 
Directrix, 56, 70 
Dirichlet, 428-430 

ref. to, xiv, 209, 339, 390, 394, 

395, 406, 415, 416, 418, 422, 

433, 462 
Dissipation of energy, 467 
Divergent parabolas, 253, 298 
Divergent series, 297, 392 
Division of the circle, 8, 316, 383, 

Diwani- numerals, 118 
Donkin, 443 
Dositheus, 46 
Dostor, 379 
Dove, 448 
D'Ovidio, 356 
Dronke, xiii 
Duality, 337, 346, 358 
Duhamel, 388, 453 
Diihring, E., xi 
Duillier, 267 
Duodecimals, 144, 147 
Dupin, 335, 336 
ref. to, 349, 366 



Duplication of the cube, 26-29, 

35, 36, 51, 57, 178 
Durege, 413 

ref. to, 360, 367 
Diirer, A., 181 
Diising, 396 
Dyck, See Groups, 367 
Dynamics, 371, 440-445 
Dziobek, xiv, 440 

Earnshaw, 447 

figure of, 299, 340 
rigidity of, 457 
size of, 249, 250 
Eddy, 349 
Edfu, 13, 61 
Edgeworth, 397 
Egyptians, 10-18, 21 
Eisenlohr, 389 
Eisenstein, 430 

ref. to, 413, 416, 426, 431, 433 
Elastic curve, 276 
Elasticity, 323, 451-457 
Electricity, 460-464 
Electro-magnetic theory of light, 

Elements (Euclid's), See Euclid, 
41-45, 70, 121, 132, 145, 
147, 148, 154, 156, 157, 159 
Elimination, See Equations, 291, 

358, 361, 385 
Elizabeth, Princess, 219 

(attraction of), 250, 322, 325, 

332, 347, 426, 441, 442 
motion of, 447 
Elliptic co-ordinates, 442 
Elliptic functions, 280, 324, 325, 
345, 383, 402, 403, 405-413, 
423, 427, 432 

Elliptic geometry, See 

Non-Euclidean geometry 
Elliptic integrals, 287, 293, 383, 

406, 408 
Ely, 434 
Encke, 427 

Energy, conservation of, 463, 465 
Enestrom, xi 
Enneper, 411 
ref. to, xiv 
Entropy, 467 

Enumerative geometry, 346 
Epicycles, 59 
Epping, ix, 9 
Equations, See Cubic equations, 

Algebra, Theory of numbers 
numerical, 170, 307, 328 
solution of, 16, 173, 177, 217, 

291, 302, 307, 323, 406 
theory of, 86, 192, 220, 224, 

251, 279, 281, 291, 382-386 
Eratosthenes, 51 

ref. to, 28, 40, 46, 82 
Errors, theory of, See Least 

Espy, 448 

Ether, luminiferous, 459 
Euclid, 40-45, 81 

ref. to, 18, 24, 25, 29, 34, 35, 

37, 38, 49, 53, 57, 61, 66, 67, 

70, 83, 84, 90, 94, 112, 121, 

125, 132, 145, 147, 158, 160, 

167, 188, 327, 353 
Euclidean space, See 

Non-Euclidean geometry 
Eudemian Summary, 18, 23, 34, 

37, 38, 40 
Eudemus, 18, 25, 52, 53, 80 
Eudoxus, 36, 37 

ref. to, 17, 32, 35, 36, 40, 42, 58 



Euler, 288-295 

ref. to, 89, 111, 209, 278, 280, 
286, 290, 300, 301, 304, 307, 
308, 311, 312, 317, 324-326, 
334, 366, 369, 390, 425, 428, 
437, 450, 451, 468 

Eulerian integrals, 325 

Eutocius, 71 

ref. to, 52, 62, 75 

Evolutes, 56, 223 

Exhaustion, method of, 30, 32, 
38, 41, 48, 196 

Exponents, 155, 176, 186, 188, 
218, 234, 280 

Factor-tables, 429 

Fagnano, 280 

Fahri des Al Karhi, 128 

Falsa positio, 107, 170 

Faraday, 464 

Favaro, xiii 

Faye, 440 

Fermat, 201, 208-212 

ref. to, 200, 201, 206, 230, 293, 
307, 308, 428 
Fermat's theorem, 209, 293 
Ferrari, 169 

ref. to, 168, 307 
Ferrel, 448 

ref. to, 438 
Ferro, Scipio, 165 
Fibonacci, See Leonardo of Pisa 
Fiedler, 349, 363, 382 
Figure of the earth, 299, 340 
Finaeus, 185 
Fine, xiii 

Finger-reckoning, 72, 137 
Finite differences, 279, 282, 293, 

314, 323, 400 
Fink, xii 

Fitzgerald, 460 

Flachenabbildung, 364 

Flamsteed, 254 

Flexure, theory of, 454 

Floridas, 165, 167 

Fluents, 238, 239 

Fluxional controversy, 263-270 

Fluxions, 232, 235-247, 388 

Focus, 56, 70, 197 

Fontaine, 293, 296 

Forbes, 463 

Force- function, See Potential, 462 

Forsyth, xii, 381, 401, 422 

Four-point problem, 397 

Fourier, 328-330 

ref. to, 203, 297, 409, 415, 428 
Fourier's series, 329, 394, 395, 

428, 451 
Fourier's theorem, 328 
Fractions, See Arithmetic 

Babylonian, 7 

continued, 184, 229, 293, 314 

decimal, 184, 185 

duodecimal, 144, 147 

Egyptian, 15 

Greek, 29, 74, 75 

Hindoo, 109 

Middle Ages, 139, 144 

Roman, 90 

sexagesimal, 7, 65, 75, 77, 147 
Franklin, 381, 434 
Frantz, xiv 
Fresnel, 458 

Fresnel's wave-surface, 243, 365 
Frezier, 333 
Fricke, 412 

Friction, theory of, 446 
Frobenius, 378, 401, 402 
Frost, 367 
Froude, 444, 448 



Fuchs, 400 

ref. to, 401, 402 functions, 403, 419 
Fuchsian groups, 402 
Functions, See Elliptic functions, 
Abelian functions, 
Hyperelliptic functions, 
Theta functions, Beta 
function, Gamma function, 
Omega function, Sigma 
function, Bessel's function, 
arbitrary, 305, 329 
definition of, 415 
theory of, 311, 313, 402, 
Funicular polygons, 348 

Gabir ben Aflah, 133 

ref. to, 147 
Galileo, 212 

ref. to, 50, 161, 187, 195, 197, 
199, 218 
Galois, 384 
Gamma function, 289 
Garbieri, 378 

Kinetic theory of, 468-471 
Gauss, 423-429 

ref. to, 89, 183, 287, 289, 292, 
307, 322, 339, 342, 351, 353, 
354, 365-367, 370, 373, 378, 
379, 383, 385, 388, 390, 399, 
406, 409, 416, 422, 435, 465 
Gauss' Analogies, 427 
Geber, See Gabir ben Aflah 
Geber's theorem, 134 
Gellibrand, 191 
Geminus, 62 

ref. to, 52, 58, 66 
Genocchi, 426 

Geodesies, 290, 442 
Geodesy, 426 

Geometry, See Curves, Surfaces, 
Curvature, Quadrature, 
Rectification, Circle 

analytic, 216-220, 222, 224, 
334, 358-367 

Arabic, 120, 124, 127, 130, 132 

Babylonian, 9 

descriptive, 333-336, 349 

Egyptian, 10-14 

Greek, 18-71, 79 

Hindoo, 112, 113 

Middle Ages, 140, 144, 147, 
148, 151, 152 

modern synthetic, 279, 
332-338, 341-358 

Renaissance, 159, 177, 179, 
184, 194 

Roman, 92 
Gerard of Cremona, 146 
Gerbert, 140-144 
Gergonne, 346 

ref. to, 207, 337 
Gerhardt, xi, 264, 268, 271 
Gerling, 427 
Germain, Sophie, 452 

ref. to, 450 
German Magnetic Union, 427 
Gerstner, 454 
Gibbs, 467 

ref. to, xii, 372 
Giovanni Campano, 148 
Girard, 193 

ref. to, 147, 187 
Glaisher, 434 

ref. to, 378, 382, 429, 437 
Glazebrook, 464 

ref. to, xv 
Gobar numerals, 94, 119 



Godfrey, 253 
Golden section, 37 
Gopel, 413 

Gordan, 364, 381, 385 
Gournerie, 349, 362 
Goursat, 400 

ref. to, 408 
Gow, ix, 40 
Graham, xii 
Grammateus, 175 
Grandi, 291 

Graphical statics, 340, 348 
Grassmann, 372-375 

ref. to, 342, 354, 369, 370, 441 
Gravitation, theory of, 248, 300, 

315, 321 
Greeks, 17-88 
Green, 461 

ref. to, 417, 447, 453, 455, 459, 
Greenhill, 413, 446 
Gregorian Calendar, 178 
Gregory, David F, 250, 331, 368 
Gregory, James, 265, 283 
Gromatici, 92 
Groups, theory of, 382-384, 

Grunert, 366 

ref. to, 373 
Gua, de, 279 
Gubar-numerals, 95, 119 
Gudermann, 412 
Guldin, 194 

ref. to, 68, 199 
Guldinus, See Guldin 
Gunter, E., 191 
Giinther, S., ix-xi, 379 
Giitzlaff, 412 

Haan, 390 

Haas, xiii 
Hachette, 335, 349 
Hadamard, 429 
Hadley, 254 
Hagen, 322 
Halifax, 156 

ref. to, 157 
Halley, 52, 248, 249, 303 
Halley's Comet, 300, 436 
Halphen, 363 

ref. to, 346, 367, 381, 401, 402, 
Halsted, x, 352 
Hamilton's numbers, 383 
Hamilton, W., 214, 368 
Hamilton, W. R., 370, 371 

ref. to, 309, 339, 340, 365, 368, 
369, 374, 378, 383, 398, 441, 
442, 458, 468 
Hammond, J, 381 
Hankel, 375 

ref. to, ix, x, 32, 108, 111, 331, 
379, 395, 422 
Hann, 449 
Hansen, 437 
Hanus, 379 
Hardy, 203 
Harkness, 422 
Harmonics, 64 
Haroun-al-Raschid, 121 
Harrington, 440 
Harriot, 193 

ref. to, 171, 177, 188, 218, 224 
Hathaway, xii 
Heat, theory of, 466-469 
Heath, 357 

Heaviside, 372, 464, 465 
Hebrews, 21 
Hegel, 435 
Heine, 396 



ref. to, 422, 434 
Helen of geometers, 217 
Helicon, 37 
Heliotrope, 423 
Helmholtz, 464 

ref. to, 354, 355, 445, 450, 459, 
463, 467, 468 
Henrici, xiv 
Henry, 463 
Heraclides, 52 
Hermite, 411 

ref. to, xiv, 383, 384, 400, 404, 
408, 414, 422, 433 
Hermotimus, 38 
Herodianic signs, 73 
Heron the Elder, 59 

ref. to, 58, 62, 75, 93, 113, 121, 
152, 163 
Herschel, J. F. W., 451 

ref. to, x, 322, 330, 415 
Hesse, 360-362 

ref. to, 344, 360, 363, 378, 384, 
385, 389, 399, 440 
Hessian, 344, 361, 381 
Heuraet, 221 
Hexagrammum mysticum, 206, 

Hicks, 445, 448 
Hilbert, 382 
Hill, 439 
Hindoos, 97-115 

ref. to, 3 
Hipparchus, 59 

ref. to, 62, 65 
Hippasus, 24 
Hippias of Elis, 28 
Hippocrates of Chios, 28, 31, 35 
Hippopede, 58 
Hirn, 467 
History of mathematics, its value, 

Hodgkinson, 454 
Holder, O., See Groups 
Holmboe, 391, 405, 408 
Homogeneity, 341, 359 
Homological figures, 206 
Honein ben Ishak, 121 
Hooke, 248 
Hoppe, 357 
Horner, 171, 385 
Hospital, T, 279 
Hoiiel, 372 
Hovarezmi, 122 

ref. to, 124, 127, 132, 145, 147 
Hudde, 220 

ref. to, 235 
Hurwitz, 418 
Hussey, 440 
Huygens, 221-223 

ref. to, 206, 212, 219, 248, 249, 
255, 272, 299, 458 
Hyde, 375 
Hydrodynamics, See Mechanics, 

277, 296, 443-448 
Hydrostatics, See Mechanics, 50, 

Hypatia, 70 

ref. to, 42 
Hyperbolic geometry, See 

Non-Euclidean geometry 
Hyperelliptic functions, 340, 382, 

406, 413, 419 
Hyperelliptic integrals, 410 
Hypergeometric series, 390, 421 
Hyperspace, 354, 355 
Hypsicles, 59 

ref. to, 7, 44, 82, 121 

Iamblichus, 84 
ref. to, 11, 24, 79 



Ibbetson, 457 
Ideal numbers, 433 
Ideler, 37 

Iehuda ben Mose Cohen, 147 
Ignoration of co-ordinates, 443 
Images, theory of, 445 
Imaginary geometry, 351 
Imaginary points, lines, etc, 347 
Imaginary quantities, 169, 193, 

280, 334, 407, 423, 434 
Imschenetzky, 399 
Incommensurables, See 

Irrationals, 42, 44, 81 
Indeterminate analysis, See 

Theory of numbers, 110, 

116, 129 
Indeterminate coefficients, 217 
Indeterminate equations, See 

Theory of numbers, 110, 

116, 128 
Indian mathematics, See Hindoos 
Indian numerals, See Arabic 

Indices, See Exponents 
Indivisibles, 197-200, 205, 224 
Induction, 396 
Infinite products, 407, 413 
Infinite series, 228, 236, 241, 256, 

288, 291, 296, 301, 313, 329, 

390-395, 406, 407, 420, 423 
Infinitesimal calculus, See 

Differential calculus 
Infinitesimals, 156, 196, 241, 242, 

Infinity, 31, 156, 196, 207, 224, 

313, 341, 354, 359 
symbol for, 224 
Insurance, 278, 396 
Integral calculus, 199, 259, 405, 

408, 429, 433 

origin of term, 275 
Interpolation, 226 
Invariant, 341, 361, 378, 382, 400, 

Inverse probability, 396 
Inverse tangents (problem of), 

197, 219, 255, 258, 259 
Involution of points, 70, 205 
Ionic School, 19-21 
Irrationals, See 

Incommensurables, 24, 29, 

79, 108, 123, 422, 434 
Irregular integrals, 401 
Ishak ben Honein, 121 
Isidorus of Seville, 137 

ref. to, 71 
Isochronous curve, 272 
Isoperimetrical figures, See 

Calculus of variations, 58, 

275, 289, 303 
Ivory, 331 

ref. to, 322 
Ivory's theorem, 332 

Jacobi, 409-411 

ref. to, 325, 339, 343, 359, 360, 
367, 378, 385, 388, 397, 405, 
407, 408, 411, 416, 426, 428, 
431, 436, 440-442, 445 
Jellet, 389 

ref. to, 445, 455 
Jerrard, 383 
Jets, 446, 451 
Jevons, 397 

Joachim, See Rhasticus 
Jochmann, 469 
John of Seville, 146, 185 
Johnson, 404 
Jordan, 384 

ref. to, 397, 400, 403 



Jordanus Nemorarius, 156 
Joubert, 412 
Joule, 466 

ref. to, 469, 471 
Julian calendar, 93 
Jurin, 274 

Kaestner, 423 

ref. to, 252 
Kant, 319, 438 

Kauffmann, See Mercator, N. 
Keill, 269, 270, 273 
Kelland, 447, 463 
Kelvin, Lord, See Thomson, W., 

ref. to, 329, 366, 417, 444-446, 
452, 457-459, 461, 465, 467, 
Kempe, 381 
Kepler, 195-197 

ref. to, 161, 181, 183, 187, 194, 
198, 202, 234, 248, 306 
Kepler's laws, 195, 247 
Kerbedz, xiv 
Ketteler, 459 
Killing, 357 
Kinckhuysen, 237 
Kinetic theory of gases, 468-471 
Kirchhoff, 463 

ref. to, 360, 445-447, 453, 459, 
Klein, 400 

ref. to, 356, 357, 360, 365, 382, 
384, 402-404, 412 
Kleinian functions, 420 
Kleinian groups, 402 
Kohlrausch, 460 
Kohn, 393 
Konig, 468 
Konigsberger, 411 

ref. to, 401, 408, 413, 414 
Kopcke, 446 
Korkine, 434 

ref. to, 397 
Korndorfer, 365 
Kowalevsky, 443 

ref. to, 402, 410, 440 
Kronig, 469 
Krause, 414 
Krazer, 414 
Kronecker, 384 

ref. to, 383, 384, 419, 426 
Kiihn, H., 369 
Kuhn, J., 254 
Kummer, 432 

ref. to, xiv, 209, 365, 393, 394, 
399, 414, 426 

Lacroix, 330, 334, 373 
Laertius, 11 
Lagrange, 303-314 

ref. to, 4, 89, 203, 209, 213, 
284, 286-288, 295, 297, 301, 
318, 323, 325, 326, 341, 345, 
354, 360, 365, 379, 423, 425, 
428, 441, 447, 450, 451, 468 
Laguerre, 356 
Lahire, de, 280 
Laisant, 372 
La Louere, 205 
Lamb, 441, 446, 447, 463 
Lambert, 300-301 

ref. to, 3, 337, 353, 365 
Lame, 454 

ref. to, 428, 454, 457 
Lame's functions, 455 
Landen, 302 

ref. to, 312, 325 
Laplace, 314-324 

ref. to, 203, 250, 279, 285, 298, 
306, 325, 332, 373, 392, 396, 



422, 423, 435, 437, 438, 448, 
450, 458, 461, 465 
Laplace's coefficients, 322 
Latitude, periodic changes in, 457 
Latus rectum, 55 
Laws of Laplace, 318 
Laws of motion, 212, 218, 247 
Least action, 294, 309, 468 
Least squares, 322, 327, 332, 423 
Lebesgue, 378, 389, 426 
Legendre, 324-327 

ref. to, 287, 293, 301, 310, 322, 
350, 407-409, 412, 425, 428 
Legendre's function, 325 
Leibniz, 254-274 

ref. to, 4, 183, 204, 233, 242, 
243, 245, 275, 280, 291-293, 
312, 367, 390, 415 
Lemoine, 397 
Lemonnier, 311 
Leodamas, 38 
Leon, 38 
Leonardo of Pisa, 148 

ref. to, 154, 159 
Leslie, x 
Le Verrier, 438 

ref. to, 439 
Levy, 349, 457 
Lewis, 445 
Lexis, 396 
Leyden jar, 463 
L'Hospital, 279 

ref. to, 267, 272 
Lie, 403 

ref. to, 398, 408 
Light, theory of, 253, 455 
Limits, method of, 246, 312 
Lindelof, 389 
Lindemann, 367 

ref. to, 3, 356, 415 

Linear associative algebra, 376 
Lintearia, 276 
Liouville, 430 

ref. to, 366, 416, 426, 431, 443 
Lipschitz, 357 

ref. to, 395, 437, 446 
Listing, 367 
Lloyd, 458 
Lobatchewsky, 350 

ref. to, 339, 352 
Local probability, 397 
Logarithmic criteria of 
convergence, 393 
Logarithmic series, 230 
Logarithms, 184, 187-192, 196, 

230, 282, 291 
Logic, 43, 368, 377, 399 
Lommel, 437, 459 
Long wave, 446 
Loomis, 448 
Lorenz, 459 
Loria, xii 
Loud, 348 

Lucas de Burgo, See Pacioli 
Ludolph, 179 
Ludolph's number, 179 
Lune, squaring of, 29 
Liiroth, 418 

ref. to, 422 

MacCullagh, 362 

ref. to, 458 
Macfarlane, 372 
Machine, arithmetical, 255, 330 
Maclaurin, 283 

ref. to, 274, 284, 326, 332, 338 
Macmahon, 381 
Magic squares, 107, 157, 280 
Magister matheseos, 158 
Main, 440 



Mainardi, 389 

Malfatti, 344, 382 
Malfatti's problem, 344, 364 
Mansion, 398 
Marie, Abbe, 324 
Marie, C. F. M., 348 
Marie, M., x, 60, 200 
Mathieu, 456 

ref. to, 412, 440, 457, 465 
Matrices, 373, 377 
Matthiessen, x 
Maudith, 156 
ref. to, 163 
Maupertius, 294, 298, 468 
Maurolycus, 177 

ref. to, 181 
Maxima and minima, 56, 202, 

217, 219, 242, 284, 389, 395, 
Maxwell, 463 

ref. to, 349, 439, 446, 455, 460, 
462, 465, 468-470 
Mayer, 465 

ref. to, 438 
McClintock, 382 
McColl, 397 
McCowan, 447 
McCullagh, 362, 459 
McMahon, 382 
Mechanics, See Dynamics, 
Hydrodynamics , 
Hydrostatics, Graphic 
statics, Laws of motion, 
Astronomy, D'Alembert's 
Bernoullis, 275, 276 
Descartes, Wallis, Wren, 

Huygens, Newton, 218, 222, 
223, 246-251 
Euler, 294 

Greek, 26, 39, 49 

Lagrange, 309 

Laplace, 319 

Leibniz, 264 

more recent work, 338, 382, 
404, 439-445, 468 

Stevin and Galileo, 184, 211 

Taylor, 282 
Meissel, 411 
Menaschmus, 36 

ref. to, 35, 38, 53, 130 
Menelaus, 64 

ref. to, 66, 182 
Mercator, G., 365 
Mercator, N., 229 

ref. to, 255 
Mere, 211 
Mersenne, 209, 223 
Mertens, 391, 429 
Meteorology, 449-450 
Method of characteristics, 346 
Method of exhaustion, 32 

ref. to, 37, 41, 48, 196 
Metius, 179 
Meunier, 366 
Meyer, A., 396, 398 
Meyer, G. F., 390 
Meyer, O. E., 446, 457, 470 
Meziriac, 208 

ref. to, 308 
Michelson, 460 
Middle Ages, 135-158 
Midorge, 202 
Minchin, 445 
Minding, 366 
Minkowsky, 432 
Mittag-Leffler, 419 
Mobius, 342 

ref. to, 342, 373, 374, 427, 438, 



Modern Europe, 160 et seq. 
Modular equations, 384, 412 
Modular functions, 412 
Mohammed ben Musa 

Hovarezmi, 122 
ref. to, 124, 127, 132, 145, 147 
Mohr, 349 
Moigno, 389 

Moivre, de, 279, 281, 284 
Mollweide, 427 
Moments in fluxionary calculus, 

Monge, 332-336 

ref. to, 288, 301, 329, 342, 349, 

366, 397 
Montmort, de, 279 
Montucla, xi, 200 
Moon, See Astronomy 
Moore, 384 
Moors, 133, 134, 144 
Moral expectation, 278 
Morley, 422 
Moschopulus, 157 
Motion, laws of, 212, 218, 247 
Mouton, 255 
Muir, xiii, 379 

Miiller, J., See Regiomontanus 
Miiller, x 

Multi-constancy, 455, 456 
Multiplication of series, 390, 392 
Musa ben Sakir, 125 
Musical proportion, 8 
Mydorge, 206 

Nachreiner, 378 

Nagelbach, 378 

Napier's rule of circular parts, 192 

Napier, J., 188, 189 

ref. to, 181, 187, 190, 191 
Napier, M., x 

Nasir Eddin, 132 

Nautical almanac, United States, 

Navier, 452 

ref. to, 446, 455 
Nebular hypothesis, 318 
Negative quantities, See Algebra, 

108, 176, 218, 298, 433 
Negative roots, See Algebra, 108, 

130, 169, 172, 176, 193 
Neil, 221 

ref. to, 230 
Neocleides, 38 
Neptune, discovery of, 438 
Nesselmann, 88 
Netto, 384 
Neumann, C, 437 

ref. to, 360, 367, 459 
Neumann, F. E., 464 

ref. to, 360, 363, 455, 457, 462, 

Newcomb, 439 

ref. to, 357, 457 
Newton, 233-254 

ref. to, 4, 58, 69, 171, 201, 216, 

222, 223, 227, 232, 277, 283, 

284, 293, 296, 298, 300, 304, 

312, 328, 332, 337, 346, 352, 

369, 385, 390, 434, 444, 450 
Newton's discovery of binomial 

theorem, 227, 228 
Newton's discovery of universal 

gravitation, 248 
Newton's parallelogram, 252 
Newton's Principia, 222, 242, 

246-250, 266, 271, 281 
Newton, controversy with 

Leibniz, 264-271 
Nicolai, 427 
Nicole, 279 



Nicolo of Brescia, See Tartaglia 
Nicomachus, 83 

ref. to, 67, 94 
Nicomedes, 58 
Nieuwentyt, 274 
Nines, casting out the, 123 
Niven, 463 
Nolan, 438 
Non-Euclidean geometry, 43, 

Nonius, 178 

ref. to, 179 
Notation, See Exponents, 

Arabic notation, 3, 84, 101, 
118, 129, 148-150, 184 

Babylonian numbers, 5-7 

decimal fractions, 186 

differential calculus, 238, 257, 
258, 303, 313, 330 

Egyptian numbers, 13 

Greek numbers, 73 

in algebra, 16, 86, 107, 155, 
172, 174, 175, 186, 193 

Roman, 90 

trigonometry, 289 
Nother, 363, 365, 384, 415 

amicable, 78, 125, 133 

defective, 78 

definitions of numbers, 434 

excessive, 78 

heteromecic, 78 

perfect, 78 

theory of numbers, 63, 87, 109, 
125, 138, 152, 207-211, 293, 
308, 326, 422-434 

triangular, 209 
Numbers of Bernoulli, 276 
Numerals, See Apices 

Arabic, 100, 118, 119, 129 
Babylonian, 4-7 
Egyptian, 14 
Greek, 73 

Oberbeck, 450 
(Enopides, 21 

ref. to, 17 
Ohm, M, 369 
Ohrtmann, x 
Olbers, 424, 435 
Oldenburg, 265 
Olivier, 349 
Omega- function, 411 
Operations, calculus of, 340 
Oppolzer, 440 
Optics, 45 
Oresme, 156 

ref. to, 186 
Orontius, 178 

Oscillation, centre of, 223, 282 
Ostrogradsky, 388, 443 
Otho, 164 
Oughtred, 194 

ref. to, 171, 187, 234 
Ovals of Descartes, 217 

it: values for 
Arabic, 125 
Archimedean, 47 
Babylonian and Hebrew, 8 
Brouncker's, 229 
Egyptian, 12 
Fagnano's, 280 
Hindoo, 114 
Leibniz's, 255 
Ludolph's, 179 
proved to be irrational, 301, 

proved to be transcendental, 1 
selection of letter 7r, 290 



Wallis', 226 
Pacioli, 157 

ref. to, 155, 165, 176, 180, 184, 
Padmanabha, 100 
Palatine anthology, 84, 138 
Pappus, 67-70 

ref. to, 40, 45, 52, 57, 58, 63, 
75, 76, 178, 207, 216 
Parabola, See Geometry, 48, 80, 

semi-cubical, 221 
Parabolic geometry, See 

Non-Euclidean geometry 
Parallelogram of forces, 213 
Parallels, 43, 327, 349, 351, 352, 

Parameter, 55 
Partial differential equations, 241, 

296, 335, 397 etseq., 442 
Partition of numbers, 433 
Pascal, 203-206 

ref. to, 207, 211, 228, 256, 280, 
330, 332, 337, 361 
Pascal's theorem, 207 
Peacock, 330 

ref. to, X, 150, 154, 187, 330, 
Pearson, 456 
Peaucellier, 380 
Peirce, B., 376 

ref. to, 339, 369, 439, 445 
Peirce, C. S., 376 

ref. to, 43, 357, 375 
Peletarius, 193 
Pell, 171, 175, 210, 255 
Pell's problem, 112, 210 
Pemberton, 234 
Pendulum, 222 
Pepin, 426 

Perier, Madame, xi 

Periodicity of functions, 406, 408 

Pernter, J. M., 450 

Perseus, 58 

Perspective, See Geometry, 206 

Perturbations, 318 

Petersen, 426 

Pfaff, 397, 398 

ref. to, 422 
Pfaffian problem, 398 
Pherecydes, 22 
Philippus, 38 
Philolaus, 25 

ref. to, 32, 78 
Philonides, 53 
Physics, mathematical, See 
Applied mathematics 
Piazzi, 435 

Picard, E., 404, 408, 420 
Picard, J., 249, 250 
Piddington, 448 
Piola, 453 
Pitiscus, 164 
Plucker, 358-360 

ref. to, 354, 359, 365 
Plana, 437, 451, 462 
Planudes, M., 157 
Plateau, 446 
Plato, 33-36 

ref. to, 4, 10, 17, 25, 36, 37, 39, 
40, 72, 78 
Plato of Tivoli, 126, 145 
Plato Tiburtinus, See Plato of 

Platonic figures, 44 
Platonic School, 33-39 
Playfair, x, 181 
Plectoidal surface, 69 
Plus and minus, signs for, 173 
Pohlke, 349 



Poincare, 400 

ref. to, xiv, 402-404, 411, 419, 

429, 448, 468 
Poinsot, 440 

ref. to, 440 
Poisson, 452 

ref. to, 203, 347, 385, 388, 409, 

437, 441, 446, 450, 452, 455, 

458, 461, 462 
Poncelet, 337, 338 

ref. to, 207, 335, 342, 356, 359, 

Poncelet 's paradox, 359 
Porisms, 45 
Porphyrius, 63 
Potential, 323, 417, 461 
Poynting, 464, 465 
Preston, 467 
Primary factors, Weierstrass' 

theory of, 412, 419 
Prime and ultimate ratios, 230, 

246, 312 
Prime numbers, 43, 51, 81, 209, 

Princess Elizabeth, 219 
Principia (Newton's), 222, 242, 

246-250, 266, 271, 281 
Pringsheim, 392-394 
Probability, 184, 211, 223, 275, 

278, 279, 285, 294, 314, 321, 

332, 396, 397 
Problem of Pappus, 69 
Problem of three bodies, 294, 297, 

Proclus, 71 

ref. to, 18, 21, 38, 40, 43, 44, 

58, 62, 67 
Progressions, first appearance of 

arithmetical and 

geometrical, 8 

Projective geometry, 358 
Proportion, 19, 25, 26, 29, 37, 42, 

44, 77, 79 
Propositiones ad acuendos 

iuvenes, 138 
Prym, 414 

Ptolemasus, See Ptolemy 
Ptolemaic System, 64 
Ptolemy, 65-67 

ref. to, 7, 9, 62, 63, 113, 120, 
122, 125, 126, 134, 160, 364 
Puiseux, 416 
Pulveriser, 110 
Purbach, 156 
ref. to, 162 
Pythagoras, 21-25, 77-80 

ref. to, 4, 17, 20, 26, 32, 42, 73, 
95, 112, 156 
Pythagorean School, 21-25 

Quadratic equations, See 

Algebra, Equations, 88, 107, 

124, 128, 130 
Quadratic reciprocity, 293, 326, 

Quadratrix, 28, 36, 68, 69 
Quadrature of curves, 48, 56, 205, 

220, 224, 256, 258 
Quadrature of the circle, See 

Circle; also see 

Circle-squarers, 7r 
Quaternions, 371 

ref. to, 369 
Quercu, a, 179 
Quetelet, 396 
ref. to, x 

Raabe, 393 
Radau, 440 
Radiometer, 470 
Rahn, 175 



Ramus, 178 
Rankine, 466 

ref. to, 467 
Rari-constancy, 455 
Ratios, 434 
Rayleigh, Lord, 451 

ref. to, 437, 447, 448, 464, 465 
Reaction polygons, 349 
Reciprocal polars, 337 
Reciprocants, 381, 421 
Recorde, 175 

ref. to, 183 
Rectification of curves, See 

Curves, 196, 206, 221, 230 
Redfield, 448 

Reductio ad absurdum, 32 
Reech, 444 
Regiomontanus, 162, 163 

ref. to, 161, 173, 177, 178, 181, 
184, 185 
Regula aurea, See Falsa positio 
Regula duorum falsorum, 123 
Regula falsa, 123 
Regular solids, 23, 36, 38, 44, 59, 

127, 195 
Reid, 448 
Reiff, xii 

Renaissance, 161-181 
Resal, 440 
Reye, 348 

ref. to, 356 
Reynolds, 447 
Rhasticus, 164 

ref. to, 161, 164 
Rheticus, See Rhasticus 
Rhind papyrus, 10-16 
Riccati, 280 

ref. to, 277 
Richard of Wallingford, 156 
Richelot, 412 

ref. to, 360, 363 
Riemann, 416-419 

ref. to, 354, 355, 364, 367, 395, 
399, 413-415, 422, 429, 448, 
457, 464 
Riemann's surfaces, 418 

ref. to, 415 
Roberts, 365 
Roberval, 199 

ref. to, 199, 217, 223 
Rolle, 280 

ref. to, 274 
Roman mathematics in Occident, 

Romans, 89-96 
Romanus, 179 

ref. to, 165, 172, 179 
Romer, 232 
Rosenberger, xv 
Rosenhain, 413 

ref. to, 412 
Roulette, 199 
Routh, 444 

ref. to, 445, 463 
Rowland, 446, 460, 464, 465 
Rudolff, 175 
Ruffini, 382 
Ruhlmann, 468 
Rule of signs, 218, 224 
Rule of three, 107, 123 

Saccheri, 353 

Sachse, xiii 

Sacro Bosco, See Halifax 

Saint- Venant, 455 

ref. to, 375, 446, 454, 459 
Salmon, xiii, 343, 362-364, 385 
Sand-counter, 76, 104 
Sarrau, 459 
Sarrus, 389 



Saturn's rings, 223, 439 
Saurin, 279 
Savart, 452 
Scaliger, 179 
Schellbach, 345 
Schepp, 422 
Schering, 357 

ref. to, 417, 426 
Schiaparelli, 37 
Schlafli, 357 

ref. to, 394, 412 
Schlegel, 376 

ref. to, XII, 357 
Schlessinger, 349 
Schlomilch, 437 
Schmidt, xiii 
Schooten, van, 220 

ref. to, 221, 234 
Schreiber, 336, 349 
Schroter, H., 365 

ref. to, 344, 411 
Schroter, J. H., 436 
Schubert, 346 
Schumacher, 427 

ref. to, 405 
Schuster, xiv 
Schwarz, 421 

ref. to, 346, 395, 402, 404, 422 
Schwarzian derivative, 421 
Scott, 378 

Screws, theory of, 441 
Secants, 164 
Sectio aurea, 37 
Section, the golden, 37 
Seeber, 433 
Segre, 356 
Seidel, 396 
Seitz, 397 
Selling, 433 
Sellmeyer, 459 

Semi-convergent series, 392 

Semi-cubical parabola, 221 

Semi- invariants, 382 

Serenus, 63 

Series, See Infinite series, 
Trigonometric series, 
Divergent series, Absolutely 
convergent series, 
Semi-convergent series, 
Fourier's series, Uniformly 
convergent series, 129, 285 

Serret, 365 

ref. to, 398, 399, 440, 443 

Servois, 331, 335, 337 

Sexagesimal system, 7, 65, 74, 77, 

Sextant, 253 

Sextus Julius Africanus, 67 

Siemens, 450 

Sigma- function, 413 

Signs, rule of, 218, 224 

Similitude (mechanical), 444 

Simony, 367 

Simplicius, 71 

Simpson, 290 

Simson, 338 
ref. to, 42, 45 

Sine, 114, 117, 126, 134, 144, 163 
origin of term, 126 

Singular solutions, 262, 308, 323 

Sluze, 220 

ref. to, 258, 260 

Smith, A., 445 

Smith, H., 430, 431 
ref. to, xiv, 411, 434 

Smith, R., 281 

Sohnke, 412 

Solid of least resistance, 250 

Solitary wave, 446 

Somoff, 445 



Sophist School, 26-32 

Sosigenes, 93 

Sound, velocity of, See Acoustics, 

314, 323 
Speidell, 192 

Spherical Harmonics, 287 
Spherical trigonometry, 64, 133, 

325, 343 
Spheroid (liquid), 448 
Spirals, 48, 69, 276 
Spitzer, 389 
Spottiswoode, 378 

ref. to, xii, 340 
Square root, 75, 108, 185 
Squaring the circle, See 

Quadrature of the circle 
Stall, 357 

Star-polygons, 24, 156, 181 
Statics, See Mechanics, 50, 212 
Statistics, 396 

Staudt, von, See Von Staudt 
Steele, 445 
Stefano, 446 
Steiner, 343, 344 

ref. to, 342, 346, 347, 359, 362, 
364, 372, 405, 416 
Stereometry, 36, 37, 44, 195 
Stern, 416, 426 
Stevin, 186 

ref. to, 155, 188, 212 
Stevinus, See Stevin 
Stewart, 338 
Stifel, 175 

ref. to, 173, 175, 180, 188 
Stirling, 284 
Stokes, 445 

ref. to, 396, 445, 447, 451, 453, 
455, 459, 465 
Story, 357 
Strassmaier, ix 

Strauch, 389 

Stringham, 357 

Strings, vibrating, 281, 296, 305 

Strutt, J. W., See Rayleigh, 451 

Struve, 427 

Sturm's theorem, 384 

Sturm, J. C. F., 385 

ref. to, 207, 328, 443, 445 
Sturm, R., 344 

St. Vincent, Gregory, 221, 229 
Substitutions, theory of, 340, 384 
Surfaces, theory of, 290, 334, 344, 

348, 361, 365 
Suter, x 

Swedenborg, 319 
Sylow, 384 

ref. to, 408 
Sylvester, 380 

ref. to, xiii, 344, 361-363, 372, 

377, 378, 382, 385, 397, 421, 

429, 432, 441, 462 
Sylvester II. (Gerbert), 138-143 
Sylvester ref. to, 252 
Symmetric functions, 291, 382, 

Synthesis, 35 
Synthetic geometry, 341-358 

Taber, 377 

Tabit ben Korra, 125 
ref. to, 121 

Tait, 330, 372, 445, 453, 467 


direct problem of, 230, 259 
in geometry, 71, 201, 216 
in trigonometry, 127, 163, 164 
inverse problem of, 197, 220, 
256, 258, 259 

Tannery, 400 
ref. to, 422 



Tartaglia, 166-168 

ref. to, 176, 177 
Tautochronous curve, 222 
Taylor's theorem, 282, 312, 314, 

388, 399 
Taylor, B., 281 

ref. to, 273, 297 
Tchebycheff, 429 
Tchirnhausen, 280 

ref. to, 260, 262, 307, 382 
Tentative assumption, See Regula 

falsa, 86, 107 
Thales, 19, 20 

ref. to, 17, 22, 24 
Theaetetus, 38 

ref. to, 40, 42, 81 
Theodoras, 80 

ref. to, 33 
Theodosius, 62 

ref. to, 125, 145, 147 
Theon of Alexandria, 70 

ref. to, 42, 59, 63, 75, 94 
Theon of Smyrna, 63, 67, 83 
Theory of equations, See 

Theory of functions, See 
Functions, 311, 313, 
401-403, 405-422 
Theory of numbers, 63, 87, 109, 
125, 138, 151, 207-211, 293, 
308, 326, 422-433 
Theory of substitutions, 384, 412 
Thermodynamics, 450, 464-468 
Theta-fuchsians, 403 
Theta- functions, 410, 411, 413, 

Theudius, 38 
Thomae, 411, 422 
Thome, 401 

ref. to, 402 

Thomson's theorem, 418 
Thomson, J., 449 
Thomson, J. J., 445 

ref. to, 462, 464 
Thomson, Sir William, See 

Kelvin (Lord), 461, 462 
ref. to, 330, 367, 417, 445, 447, 

453, 457, 459-461, 466, 467, 

Three bodies, problem of, 294, 

297, 439 
Thymaridas, 84 
Tides, 323, 447 
Timasus of Locri, 33 
Tisserand, 440 
Todhunter, 389 

ref. to, xi, 437 
Tonstall, 184 
Torricelli, 199 
Trajectories, 272, 277 
Triangulum characteristicum, 256 
Trigonometric series, See 

Fourier's series, 329, 395, 

Trigonometry, 59, 65, 114-115, 

126, 127, 133, 156, 162, 163, 

179, 185, 186, 191, 277, 282, 

285, 290, 301 
spherical, 66, 133, 325, 343 
Trisection of angles, 26, 35, 57, 

Trochoid, 199 
Trouton, 460 
Trudi, 378 
Tucker, xiv 

Twisted Cartesian, 363 
Tycho Brahe, 127, 161, 195 

Ubaldo, 213 

Ultimate multiplier, theory of, 



Ulug Beg, 132 

Undulatory theory of light, 223, 

395, 442, 457-459 
Universities of Cologne, Leipzig, 

Oxford, Paris, and Prague, 


Valson, xiii 

Van Ceulen, See Ludolph 

Vandermonde, 323 

ref. to, 307, 323 
Van Schooten, 220 

ref. to, 221, 234 
Variation of arbitrary consonants, 

Varignon, 279 

ref. to, 275 
Varying action, principle of, 340, 

371, 442 
Venturi, 60 
Veronese, 356 

ref. to, 357 
Versed sine, 114 
Vibrating rods, 451 
Vibrating strings, 281, 296, 305 
Vicat, 454 

ref. to, 455 
Victorius, 91 
Vieta, 170 

ref. to, 57, 165, 176, 178, 179, 
194, 228, 234, 252, 307 
Vincent, Gregory St., 221, 229 
Virtual velocities, 39, 309 
Viviani, 200 
Vlacq, 191 
Voigt, xiv, 426, 459 
Volaria, 276 

Von Helmholtz, See Helmholtz 
Von Staudt, 347, 348 

ref. to, 340, 342, 344 

Vortex motion, 446 
Vortex rings, 445 
Voss, 357 
ref. to, 392 

Waldo, 449 
Walker, 376 
Wallis, 223-226 

ref. to, 113, 187, 205, 208, 218, 
219, 229, 234, 267 
Waltershausen, xii 
Wand, 467 
Wantzel, 382 
Warring, 307, 384 
Watson, J. C, 440 
Watson, S., 397 
Wave theory, See Undulatory 

Waves, 446-450 
Weber, H. H., 414 
Weber, W. E., 460 

ref. to, 416, 423, 453, 463, 464 
Weierstrass, 419 

ref. to, 382, 395, 411-413, 419, 
421, 422, 434 
Weigel, 255 
Weiler, 397 
Werner, 177 
Wertheim, 455 
Westergaard, 396 
Wheatstone, 450 
Whewell, ix, 49, 294 
Whiston, 251 
Whitney, 101 
Widmann, 175 
Wiener, xii 
Williams, 311 
Wilson, 308 
Wilson's theorem, 308 
Winds, 448-450 



Winkler, 457 
Witch of Agnesi, 302 
Wittstein, xii 
Woepcke, 96, 119 
Wolf, C, 281 

ref. to, 194 
Wolf, R., xii 
Wolstenholme, 397 
Woodhouse, 389 
Wren, 206 

ref. to, 219, 230, 248, 334 
Wronski, 378 

Xenocrates, 33 
Xylander, 177 

Young, 458 

ref. to, 450, 451 

Zag, 147 
Zahn, xiii 
Zehfuss, 378 
Zeller, 426 
Zeno, 31 
Zenodorus, 58 

(symbol for), 7, 101 

origin of term, 149 
Zeuthen, 365 

ref. to, ix, 346 
Zeuxippus, 46 
Zolotareff, 433 

ref. to, 434 



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